Permutations¶
The Permutations module. Use Permutation?
to get information about
the Permutation class, and Permutations?
to get information about
the combinatorial class of permutations.
Warning
This file defined Permutation
which depends upon
CombinatorialElement
despite it being deprecated (see
Issue #13742). This is dangerous. In particular, the
Permutation._left_to_right_multiply_on_right()
method (which can
be called through multiplication) disables the input checks (see
Permutation()
). This should not happen. Do not trust the results.
What does this file define ?¶
The main part of this file consists in the definition of permutation objects,
i.e. the Permutation()
method and the
Permutation
class. Global options for
elements of the permutation class can be set through the
Permutations.options()
object.
Below are listed all methods and classes defined in this file.
Methods of Permutations objects
Return the product of |
|
Return the product of |
|
Return the size of the permutation |
|
Return the disjoint-cycles representation of |
|
Return the permutation that follows |
|
Return the permutation that comes directly before |
|
Return a tableau of shape |
|
Return the permutation |
|
Return |
|
Return a |
|
Return the signature of the permutation |
|
Return |
|
Return a matrix representing the permutation |
|
Return the rank of |
|
Return the inversion vector of a permutation |
|
Return a list of the inversions of permutation |
|
Return the permutation obtained by sorting |
|
Return a digraph representation of |
|
Display the permutation as a drawing. |
|
Return the number of inversions in the permutation |
|
Return the |
|
Return the number of |
|
Return the Coxeter length of a permutation |
|
Return the inverse of a permutation |
|
Return the |
|
Return the |
|
Return a list of the runs in the permutation |
|
Return the length of the longest increasing subsequences of |
|
Return the list of the longest increasing subsequences of |
|
Return the number of longest increasing subsequences |
|
Return the cycle type of |
|
Return the image of the permutation |
|
Return the image of the permutation |
|
Return the image of the permutation |
|
Return the image of the permutation |
|
Return destandardization of |
|
Return the Lehmer code of the permutation |
|
Return the Lehmer cocode of |
|
Return the reduced word of the permutation |
|
Return a list of the reduced words of the permutation |
|
An iterator for the reduced words of the permutation |
|
Return a lexicographically minimal reduced word of a permutation |
|
Return a list of the fixed points of the permutation |
|
Return |
|
Return |
|
Return the number of fixed points of the permutation |
|
Return the list of the positions of the recoils of the permutation |
|
Return the number of recoils of the permutation |
|
Return the composition corresponding to the recoils of |
|
Return the list of the descents of the permutation |
|
Return a list of the idescents of |
|
Return the list obtained by mapping each position in |
|
Return the number of descents of the permutation |
|
Return the number of idescents of the permutation |
|
Return the composition corresponding to the descents of |
|
Return the descent polynomial of the permutation |
|
Return the major index of the permutation |
|
Return the inverse major index of the permutation |
|
Return the major code of the permutation |
|
Return a list of the peaks of the permutation |
|
Return the number of peaks of the permutation |
|
Return a list of the saliances of the permutation |
|
Return the number of saliances of the permutation |
|
Return |
|
Return all the numbers |
|
Return the list of inversions of |
|
Return an iterator over Bruhat inversions of |
|
Return a list of the permutations covering |
|
An iterator for the permutations covering |
|
Return a list of the permutations covered by |
|
An iterator for the permutations covered by |
|
Return the combinatorial class of permutations smaller than or equal to |
|
Return the combinatorial class of permutations greater than or equal to |
|
Return |
|
Return a list of the permutations covering |
|
Return a list of the permutations covered by |
|
Return a list of permutations smaller than or equal to |
|
Return a list of permutations greater than or equal to |
|
Return an iterator over permutations in an interval of the permutohedron order. |
|
Return a list of permutations in an interval of the permutohedron order. |
|
Test whether the permutation |
|
Test whether the permutation |
|
Return the list of positions where the pattern |
|
Return the permutation obtained by reversing the 1-line notation of |
|
Return the complement of the permutation which is obtained by replacing each value \(x\) in the 1-line notation of |
|
Return the permutation poset of |
|
Return a dictionary corresponding to the permutation |
|
Return the action of the permutation |
|
Return the pair of standard tableaux obtained by running the Robinson-Schensted Algorithm on |
|
Return the left standard tableau after performing the RSK algorithm. |
|
Return the right standard tableau after performing the RSK algorithm. |
|
Return the increasing tree of |
|
Return the shape of the increasing tree of |
|
Return the binary search tree of |
|
Iterate over the equivalence class of |
|
Return the shape of the tableaux obtained by the RSK algorithm. |
|
Return the permutation obtained by removing any fixed points at the end of |
|
Return the plain retract of |
|
Return the direct-product retract of |
|
Return the Okounkov-Vershik retract of |
|
Return the coset-type of |
|
Return the shape of the binary search tree of |
|
Return the right (or left) shifted concatenation of |
|
Return the shifted shuffle of |
Other classes defined in this file
Functions defined in this file
Return the permutation corresponding to major code |
|
Return a Permutation give a |
|
Return the permutation with the specified lexicographic rank. |
|
Return the permutation corresponding to inversion vector |
|
Return the permutation with given disjoint-cycle representation |
|
Return the permutation with Lehmer code |
|
Return the permutation corresponding to the reduced word |
|
Return a given bistochastic matrix as a nonnegative linear combination of permutations. |
|
Return a partial permutation representing the bounded affine permutation of a matrix. |
|
Return a list of all the permutations in a given descent class (i. e., having a given descents composition). |
|
Return the smallest element of a descent class. |
|
Return the largest element of a descent class. |
|
Return |
|
Return |
|
Return a standard permutation corresponding to the permutation |
AUTHORS:
Mike Hansen
Dan Drake (2008-04-07): allow Permutation() to take lists of tuples
Sébastien Labbé (2009-03-17): added robinson_schensted_inverse
Travis Scrimshaw:
(2012-08-16):
to_standard()
no longer modifies input(2013-01-19): Removed RSK implementation and moved to
rsk
.(2013-07-13): Removed
CombinatorialClass
and moved permutations to the category framework.
Darij Grinberg (2013-09-07): added methods; ameliorated Issue #14885 by exposing and documenting methods for global-independent multiplication.
Travis Scrimshaw (2014-02-05): Made
StandardPermutations_n
a finite Weyl group to make it more uniform withSymmetricGroup
. Added ability to compute the conjugacy classes.Trevor K. Karn (2022-08-05): Add
Permutation.n_reduced_words()
Amrutha P, Shriya M, Divya Aggarwal (2022-08-16): Added Multimajor Index.
Classes and methods¶
- class sage.combinat.permutation.Arrangements[source]¶
Bases:
Permutations
An arrangement of a multiset
mset
is an ordered selection without repetitions. It is represented by a list that contains only elements frommset
, but maybe in a different order.Arrangements
returns the combinatorial class of arrangements of the multisetmset
that containk
elements.EXAMPLES:
sage: mset = [1,1,2,3,4,4,5] sage: Arrangements(mset, 2).list() # needs sage.libs.gap [[1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 1], [2, 3], [2, 4], [2, 5], [3, 1], [3, 2], [3, 4], [3, 5], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 1], [5, 2], [5, 3], [5, 4]] sage: Arrangements(mset, 2).cardinality() # needs sage.libs.gap 22 sage: Arrangements( ["c","a","t"], 2 ).list() [['c', 'a'], ['c', 't'], ['a', 'c'], ['a', 't'], ['t', 'c'], ['t', 'a']] sage: Arrangements( ["c","a","t"], 3 ).list() [['c', 'a', 't'], ['c', 't', 'a'], ['a', 'c', 't'], ['a', 't', 'c'], ['t', 'c', 'a'], ['t', 'a', 'c']]
>>> from sage.all import * >>> mset = [Integer(1),Integer(1),Integer(2),Integer(3),Integer(4),Integer(4),Integer(5)] >>> Arrangements(mset, Integer(2)).list() # needs sage.libs.gap [[1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 1], [2, 3], [2, 4], [2, 5], [3, 1], [3, 2], [3, 4], [3, 5], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 1], [5, 2], [5, 3], [5, 4]] >>> Arrangements(mset, Integer(2)).cardinality() # needs sage.libs.gap 22 >>> Arrangements( ["c","a","t"], Integer(2) ).list() [['c', 'a'], ['c', 't'], ['a', 'c'], ['a', 't'], ['t', 'c'], ['t', 'a']] >>> Arrangements( ["c","a","t"], Integer(3) ).list() [['c', 'a', 't'], ['c', 't', 'a'], ['a', 'c', 't'], ['a', 't', 'c'], ['t', 'c', 'a'], ['t', 'a', 'c']]
- cardinality()[source]¶
Return the cardinality of
self
.EXAMPLES:
sage: A = Arrangements([1,1,2,3,4,4,5], 2) sage: A.cardinality() # needs sage.libs.gap 22
>>> from sage.all import * >>> A = Arrangements([Integer(1),Integer(1),Integer(2),Integer(3),Integer(4),Integer(4),Integer(5)], Integer(2)) >>> A.cardinality() # needs sage.libs.gap 22
- class sage.combinat.permutation.Arrangements_msetk(mset, k)[source]¶
Bases:
Arrangements
,Permutations_msetk
Arrangements of length \(k\) of a multiset \(M\).
- class sage.combinat.permutation.Arrangements_setk(s, k)[source]¶
Bases:
Arrangements
,Permutations_setk
Arrangements of length \(k\) of a set \(S\).
- class sage.combinat.permutation.CyclicPermutations(mset)[source]¶
Bases:
Permutations_mset
Return the class of all cyclic permutations of
mset
in cycle notation. These are the same as necklaces.INPUT:
mset
– a multiset
EXAMPLES:
sage: CyclicPermutations(range(4)).list() # needs sage.combinat [[0, 1, 2, 3], [0, 1, 3, 2], [0, 2, 1, 3], [0, 2, 3, 1], [0, 3, 1, 2], [0, 3, 2, 1]] sage: CyclicPermutations([1,1,1]).list() # needs sage.combinat [[1, 1, 1]]
>>> from sage.all import * >>> CyclicPermutations(range(Integer(4))).list() # needs sage.combinat [[0, 1, 2, 3], [0, 1, 3, 2], [0, 2, 1, 3], [0, 2, 3, 1], [0, 3, 1, 2], [0, 3, 2, 1]] >>> CyclicPermutations([Integer(1),Integer(1),Integer(1)]).list() # needs sage.combinat [[1, 1, 1]]
- iterator(distinct=False)[source]¶
EXAMPLES:
sage: CyclicPermutations(range(4)).list() # indirect doctest # needs sage.combinat [[0, 1, 2, 3], [0, 1, 3, 2], [0, 2, 1, 3], [0, 2, 3, 1], [0, 3, 1, 2], [0, 3, 2, 1]] sage: CyclicPermutations([1,1,1]).list() # needs sage.combinat [[1, 1, 1]] sage: CyclicPermutations([1,1,1]).list(distinct=True) # needs sage.combinat [[1, 1, 1], [1, 1, 1]]
>>> from sage.all import * >>> CyclicPermutations(range(Integer(4))).list() # indirect doctest # needs sage.combinat [[0, 1, 2, 3], [0, 1, 3, 2], [0, 2, 1, 3], [0, 2, 3, 1], [0, 3, 1, 2], [0, 3, 2, 1]] >>> CyclicPermutations([Integer(1),Integer(1),Integer(1)]).list() # needs sage.combinat [[1, 1, 1]] >>> CyclicPermutations([Integer(1),Integer(1),Integer(1)]).list(distinct=True) # needs sage.combinat [[1, 1, 1], [1, 1, 1]]
- list(distinct=False)[source]¶
EXAMPLES:
sage: CyclicPermutations(range(4)).list() # needs sage.combinat [[0, 1, 2, 3], [0, 1, 3, 2], [0, 2, 1, 3], [0, 2, 3, 1], [0, 3, 1, 2], [0, 3, 2, 1]]
>>> from sage.all import * >>> CyclicPermutations(range(Integer(4))).list() # needs sage.combinat [[0, 1, 2, 3], [0, 1, 3, 2], [0, 2, 1, 3], [0, 2, 3, 1], [0, 3, 1, 2], [0, 3, 2, 1]]
- class sage.combinat.permutation.CyclicPermutationsOfPartition(partition)[source]¶
Bases:
Permutations
Combinations of cyclic permutations of each cell of a given partition.
This is the same as a Cartesian product of necklaces.
EXAMPLES:
sage: CyclicPermutationsOfPartition([[1,2,3,4],[5,6,7]]).list() # needs sage.combinat [[[1, 2, 3, 4], [5, 6, 7]], [[1, 2, 4, 3], [5, 6, 7]], [[1, 3, 2, 4], [5, 6, 7]], [[1, 3, 4, 2], [5, 6, 7]], [[1, 4, 2, 3], [5, 6, 7]], [[1, 4, 3, 2], [5, 6, 7]], [[1, 2, 3, 4], [5, 7, 6]], [[1, 2, 4, 3], [5, 7, 6]], [[1, 3, 2, 4], [5, 7, 6]], [[1, 3, 4, 2], [5, 7, 6]], [[1, 4, 2, 3], [5, 7, 6]], [[1, 4, 3, 2], [5, 7, 6]]]
>>> from sage.all import * >>> CyclicPermutationsOfPartition([[Integer(1),Integer(2),Integer(3),Integer(4)],[Integer(5),Integer(6),Integer(7)]]).list() # needs sage.combinat [[[1, 2, 3, 4], [5, 6, 7]], [[1, 2, 4, 3], [5, 6, 7]], [[1, 3, 2, 4], [5, 6, 7]], [[1, 3, 4, 2], [5, 6, 7]], [[1, 4, 2, 3], [5, 6, 7]], [[1, 4, 3, 2], [5, 6, 7]], [[1, 2, 3, 4], [5, 7, 6]], [[1, 2, 4, 3], [5, 7, 6]], [[1, 3, 2, 4], [5, 7, 6]], [[1, 3, 4, 2], [5, 7, 6]], [[1, 4, 2, 3], [5, 7, 6]], [[1, 4, 3, 2], [5, 7, 6]]]
sage: CyclicPermutationsOfPartition([[1,2,3,4],[4,4,4]]).list() # needs sage.combinat [[[1, 2, 3, 4], [4, 4, 4]], [[1, 2, 4, 3], [4, 4, 4]], [[1, 3, 2, 4], [4, 4, 4]], [[1, 3, 4, 2], [4, 4, 4]], [[1, 4, 2, 3], [4, 4, 4]], [[1, 4, 3, 2], [4, 4, 4]]]
>>> from sage.all import * >>> CyclicPermutationsOfPartition([[Integer(1),Integer(2),Integer(3),Integer(4)],[Integer(4),Integer(4),Integer(4)]]).list() # needs sage.combinat [[[1, 2, 3, 4], [4, 4, 4]], [[1, 2, 4, 3], [4, 4, 4]], [[1, 3, 2, 4], [4, 4, 4]], [[1, 3, 4, 2], [4, 4, 4]], [[1, 4, 2, 3], [4, 4, 4]], [[1, 4, 3, 2], [4, 4, 4]]]
sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list() # needs sage.combinat [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]
>>> from sage.all import * >>> CyclicPermutationsOfPartition([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(4),Integer(4)]]).list() # needs sage.combinat [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]
sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list(distinct=True) # needs sage.combinat [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]], [[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]
>>> from sage.all import * >>> CyclicPermutationsOfPartition([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(4),Integer(4)]]).list(distinct=True) # needs sage.combinat [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]], [[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]
- class Element[source]¶
Bases:
ClonableArray
A cyclic permutation of a partition.
- check()[source]¶
Check that
self
is a valid element.EXAMPLES:
sage: CP = CyclicPermutationsOfPartition([[1,2,3,4],[5,6,7]]) sage: elt = CP[0] # needs sage.combinat sage: elt.check() # needs sage.combinat
>>> from sage.all import * >>> CP = CyclicPermutationsOfPartition([[Integer(1),Integer(2),Integer(3),Integer(4)],[Integer(5),Integer(6),Integer(7)]]) >>> elt = CP[Integer(0)] # needs sage.combinat >>> elt.check() # needs sage.combinat
- iterator(distinct=False)[source]¶
AUTHORS:
Robert Miller
EXAMPLES:
sage: CyclicPermutationsOfPartition([[1,2,3,4], # indirect doctest # needs sage.combinat ....: [5,6,7]]).list() [[[1, 2, 3, 4], [5, 6, 7]], [[1, 2, 4, 3], [5, 6, 7]], [[1, 3, 2, 4], [5, 6, 7]], [[1, 3, 4, 2], [5, 6, 7]], [[1, 4, 2, 3], [5, 6, 7]], [[1, 4, 3, 2], [5, 6, 7]], [[1, 2, 3, 4], [5, 7, 6]], [[1, 2, 4, 3], [5, 7, 6]], [[1, 3, 2, 4], [5, 7, 6]], [[1, 3, 4, 2], [5, 7, 6]], [[1, 4, 2, 3], [5, 7, 6]], [[1, 4, 3, 2], [5, 7, 6]]]
>>> from sage.all import * >>> CyclicPermutationsOfPartition([[Integer(1),Integer(2),Integer(3),Integer(4)], # indirect doctest # needs sage.combinat ... [Integer(5),Integer(6),Integer(7)]]).list() [[[1, 2, 3, 4], [5, 6, 7]], [[1, 2, 4, 3], [5, 6, 7]], [[1, 3, 2, 4], [5, 6, 7]], [[1, 3, 4, 2], [5, 6, 7]], [[1, 4, 2, 3], [5, 6, 7]], [[1, 4, 3, 2], [5, 6, 7]], [[1, 2, 3, 4], [5, 7, 6]], [[1, 2, 4, 3], [5, 7, 6]], [[1, 3, 2, 4], [5, 7, 6]], [[1, 3, 4, 2], [5, 7, 6]], [[1, 4, 2, 3], [5, 7, 6]], [[1, 4, 3, 2], [5, 7, 6]]]
sage: CyclicPermutationsOfPartition([[1,2,3,4],[4,4,4]]).list() # needs sage.combinat [[[1, 2, 3, 4], [4, 4, 4]], [[1, 2, 4, 3], [4, 4, 4]], [[1, 3, 2, 4], [4, 4, 4]], [[1, 3, 4, 2], [4, 4, 4]], [[1, 4, 2, 3], [4, 4, 4]], [[1, 4, 3, 2], [4, 4, 4]]]
>>> from sage.all import * >>> CyclicPermutationsOfPartition([[Integer(1),Integer(2),Integer(3),Integer(4)],[Integer(4),Integer(4),Integer(4)]]).list() # needs sage.combinat [[[1, 2, 3, 4], [4, 4, 4]], [[1, 2, 4, 3], [4, 4, 4]], [[1, 3, 2, 4], [4, 4, 4]], [[1, 3, 4, 2], [4, 4, 4]], [[1, 4, 2, 3], [4, 4, 4]], [[1, 4, 3, 2], [4, 4, 4]]]
sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list() # needs sage.combinat [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]
>>> from sage.all import * >>> CyclicPermutationsOfPartition([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(4),Integer(4)]]).list() # needs sage.combinat [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]
sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list(distinct=True) # needs sage.combinat [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]], [[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]
>>> from sage.all import * >>> CyclicPermutationsOfPartition([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(4),Integer(4)]]).list(distinct=True) # needs sage.combinat [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]], [[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]
- list(distinct=False)[source]¶
EXAMPLES:
sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list() # needs sage.combinat [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]] sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list(distinct=True) # needs sage.combinat [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]], [[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]
>>> from sage.all import * >>> CyclicPermutationsOfPartition([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(4),Integer(4)]]).list() # needs sage.combinat [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]] >>> CyclicPermutationsOfPartition([[Integer(1),Integer(2),Integer(3)],[Integer(4),Integer(4),Integer(4)]]).list(distinct=True) # needs sage.combinat [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]], [[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]
- class sage.combinat.permutation.PatternAvoider(parent, patterns)[source]¶
Bases:
GenericBacktracker
EXAMPLES:
sage: from sage.combinat.permutation import PatternAvoider sage: P = Permutations(4) sage: p = PatternAvoider(P, [[1,2,3]]) sage: loads(dumps(p)) <sage.combinat.permutation.PatternAvoider object at 0x...>
>>> from sage.all import * >>> from sage.combinat.permutation import PatternAvoider >>> P = Permutations(Integer(4)) >>> p = PatternAvoider(P, [[Integer(1),Integer(2),Integer(3)]]) >>> loads(dumps(p)) <sage.combinat.permutation.PatternAvoider object at 0x...>
- class sage.combinat.permutation.Permutation(parent, l, algorithm='lex', sjt=None, check=True)[source]¶
Bases:
CombinatorialElement
A permutation.
Converts
l
to a permutation on \(\{1, 2, \ldots, n\}\).INPUT:
l
– can be any one of the following:an instance of
Permutation
,list of integers, viewed as one-line permutation notation. The construction checks that you give an acceptable entry. To avoid the check, use the
check
option.string, expressing the permutation in cycle notation.
list of tuples of integers, expressing the permutation in cycle notation.
a
PermutationGroupElement
a pair of two standard tableaux of the same shape. This yields the permutation obtained from the pair using the inverse of the Robinson-Schensted algorithm.
check
– boolean (default:True
); whether to check that input is correct; slows the function down, but ensures that nothing bad happensalgorithm
– string (default:'lex'
); the algorithm used to generate the permutations. Supported algorithms are:'lex'
: lexicographic order generation, this is the default algorithm'sjt'
: Steinhaus-Johnson-Trotter algorithm to generate permutations using only transposition of two elements in the list. It is highly recommended to setcheck=True
(default value).
sjt
– SJT (default:None
); theSJT
object holding the permutation internal state. This should only be specified when initializing with non-identity permutation.
Warning
Since Issue #13742 the input is checked for correctness : it is not accepted unless it actually is a permutation on \(\{1, \ldots, n\}\). It means that some
Permutation()
objects cannot be created anymore without settingcheck=False
, as there is no certainty that its functions can handle them, and this should be fixed in a much better way ASAP (the functions should be rewritten to handle those cases, and new tests be added).Warning
There are two possible conventions for multiplying permutations, and the one currently enabled in Sage by default is the one which has \((pq)(i) = q(p(i))\) for any permutations \(p \in S_n\) and \(q \in S_n\) and any \(1 \leq i \leq n\). (This equation looks less strange when the action of permutations on numbers is written from the right: then it takes the form \(i^{pq} = (i^p)^q\), which is an associativity law). There is an alternative convention, which has \((pq)(i) = p(q(i))\) instead. The conventions can be switched at runtime using
sage.combinat.permutation.Permutations.options()
. It is best for code not to rely on this setting being set to a particular standard, but rather use the methodsleft_action_product()
andright_action_product()
for multiplying permutations (these methods don’t depend on the setting). See Issue #14885 for more details.Note
The
bruhat*
methods refer to the strong Bruhat order. To use the weak Bruhat order, look underpermutohedron*
.EXAMPLES:
sage: Permutation([2,1]) [2, 1] sage: Permutation([2, 1, 4, 5, 3]) [2, 1, 4, 5, 3] sage: Permutation('(1,2)') [2, 1] sage: Permutation('(1,2)(3,4,5)') [2, 1, 4, 5, 3] sage: Permutation( ((1,2),(3,4,5)) ) [2, 1, 4, 5, 3] sage: Permutation( [(1,2),(3,4,5)] ) [2, 1, 4, 5, 3] sage: Permutation( ((1,2)) ) [2, 1] sage: Permutation( (1,2) ) [2, 1] sage: Permutation( ((1,2),) ) [2, 1] sage: Permutation( ((1,),) ) [1] sage: Permutation( (1,) ) [1] sage: Permutation( () ) [] sage: Permutation( ((),) ) [] sage: p = Permutation((1, 2, 5)); p [2, 5, 3, 4, 1] sage: type(p) <class 'sage.combinat.permutation.StandardPermutations_n_with_category.element_class'>
>>> from sage.all import * >>> Permutation([Integer(2),Integer(1)]) [2, 1] >>> Permutation([Integer(2), Integer(1), Integer(4), Integer(5), Integer(3)]) [2, 1, 4, 5, 3] >>> Permutation('(1,2)') [2, 1] >>> Permutation('(1,2)(3,4,5)') [2, 1, 4, 5, 3] >>> Permutation( ((Integer(1),Integer(2)),(Integer(3),Integer(4),Integer(5))) ) [2, 1, 4, 5, 3] >>> Permutation( [(Integer(1),Integer(2)),(Integer(3),Integer(4),Integer(5))] ) [2, 1, 4, 5, 3] >>> Permutation( ((Integer(1),Integer(2))) ) [2, 1] >>> Permutation( (Integer(1),Integer(2)) ) [2, 1] >>> Permutation( ((Integer(1),Integer(2)),) ) [2, 1] >>> Permutation( ((Integer(1),),) ) [1] >>> Permutation( (Integer(1),) ) [1] >>> Permutation( () ) [] >>> Permutation( ((),) ) [] >>> p = Permutation((Integer(1), Integer(2), Integer(5))); p [2, 5, 3, 4, 1] >>> type(p) <class 'sage.combinat.permutation.StandardPermutations_n_with_category.element_class'>
Generate permutations using the Steinhaus-Johnson Trotter algorithm. The output is not in lexicographic order:
sage: p = Permutation([1, 2, 3, 4], algorithm='sjt'); p [1, 2, 3, 4] sage: p = p.next(); p [1, 2, 4, 3] sage: p = p.next(); p [1, 4, 2, 3] sage: p = Permutation([1, 2, 3], algorithm='sjt') sage: for _ in range(6): ....: p = p.next() sage: p False sage: Permutation([1, 3, 2, 4], algorithm='sjt') Traceback (most recent call last): ... ValueError: no internal state directions were given for non-identity starting permutation for Steinhaus-Johnson-Trotter algorithm
>>> from sage.all import * >>> p = Permutation([Integer(1), Integer(2), Integer(3), Integer(4)], algorithm='sjt'); p [1, 2, 3, 4] >>> p = p.next(); p [1, 2, 4, 3] >>> p = p.next(); p [1, 4, 2, 3] >>> p = Permutation([Integer(1), Integer(2), Integer(3)], algorithm='sjt') >>> for _ in range(Integer(6)): ... p = p.next() >>> p False >>> Permutation([Integer(1), Integer(3), Integer(2), Integer(4)], algorithm='sjt') Traceback (most recent call last): ... ValueError: no internal state directions were given for non-identity starting permutation for Steinhaus-Johnson-Trotter algorithm
Construction from a string in cycle notation:
sage: p = Permutation( '(4,5)' ); p [1, 2, 3, 5, 4]
>>> from sage.all import * >>> p = Permutation( '(4,5)' ); p [1, 2, 3, 5, 4]
The size of the permutation is the maximum integer appearing; add a 1-cycle to increase this:
sage: p2 = Permutation( '(4,5)(10)' ); p2 [1, 2, 3, 5, 4, 6, 7, 8, 9, 10] sage: len(p); len(p2) 5 10
>>> from sage.all import * >>> p2 = Permutation( '(4,5)(10)' ); p2 [1, 2, 3, 5, 4, 6, 7, 8, 9, 10] >>> len(p); len(p2) 5 10
We construct a
Permutation
from aPermutationGroupElement
:sage: g = PermutationGroupElement([2,1,3]) # needs sage.groups sage: Permutation(g) # needs sage.groups [2, 1, 3]
>>> from sage.all import * >>> g = PermutationGroupElement([Integer(2),Integer(1),Integer(3)]) # needs sage.groups >>> Permutation(g) # needs sage.groups [2, 1, 3]
From a pair of tableaux of the same shape. This uses the inverse of the Robinson-Schensted algorithm:
sage: # needs sage.combinat sage: p = [[1, 4, 7], [2, 5], [3], [6]] sage: q = [[1, 2, 5], [3, 6], [4], [7]] sage: P = Tableau(p) sage: Q = Tableau(q) sage: Permutation( (p, q) ) [3, 6, 5, 2, 7, 4, 1] sage: Permutation( [p, q] ) [3, 6, 5, 2, 7, 4, 1] sage: Permutation( (P, Q) ) [3, 6, 5, 2, 7, 4, 1] sage: Permutation( [P, Q] ) [3, 6, 5, 2, 7, 4, 1]
>>> from sage.all import * >>> # needs sage.combinat >>> p = [[Integer(1), Integer(4), Integer(7)], [Integer(2), Integer(5)], [Integer(3)], [Integer(6)]] >>> q = [[Integer(1), Integer(2), Integer(5)], [Integer(3), Integer(6)], [Integer(4)], [Integer(7)]] >>> P = Tableau(p) >>> Q = Tableau(q) >>> Permutation( (p, q) ) [3, 6, 5, 2, 7, 4, 1] >>> Permutation( [p, q] ) [3, 6, 5, 2, 7, 4, 1] >>> Permutation( (P, Q) ) [3, 6, 5, 2, 7, 4, 1] >>> Permutation( [P, Q] ) [3, 6, 5, 2, 7, 4, 1]
- RS_partition()[source]¶
Return the shape of the tableaux obtained by applying the RSK algorithm to
self
.EXAMPLES:
sage: Permutation([1,4,3,2]).RS_partition() # needs sage.combinat [2, 1, 1]
>>> from sage.all import * >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).RS_partition() # needs sage.combinat [2, 1, 1]
- absolute_length()[source]¶
Return the absolute length of
self
The absolute length is the length of the shortest expression of the element as a product of reflections.
For permutations in the symmetric groups, the absolute length is the size minus the number of its disjoint cycles.
EXAMPLES:
sage: Permutation([4,2,3,1]).absolute_length() # needs sage.combinat 1
>>> from sage.all import * >>> Permutation([Integer(4),Integer(2),Integer(3),Integer(1)]).absolute_length() # needs sage.combinat 1
- action(a)[source]¶
Return the action of the permutation
self
on a lista
.The action of a permutation \(p \in S_n\) on an \(n\)-element list \((a_1, a_2, \ldots, a_n)\) is defined to be \((a_{p(1)}, a_{p(2)}, \ldots, a_{p(n)})\).
EXAMPLES:
sage: p = Permutation([2,1,3]) sage: a = list(range(3)) sage: p.action(a) [1, 0, 2] sage: b = [1,2,3,4] sage: p.action(b) Traceback (most recent call last): ... ValueError: len(a) must equal len(self) sage: q = Permutation([2,3,1]) sage: a = list(range(3)) sage: q.action(a) [1, 2, 0]
>>> from sage.all import * >>> p = Permutation([Integer(2),Integer(1),Integer(3)]) >>> a = list(range(Integer(3))) >>> p.action(a) [1, 0, 2] >>> b = [Integer(1),Integer(2),Integer(3),Integer(4)] >>> p.action(b) Traceback (most recent call last): ... ValueError: len(a) must equal len(self) >>> q = Permutation([Integer(2),Integer(3),Integer(1)]) >>> a = list(range(Integer(3))) >>> q.action(a) [1, 2, 0]
- avoids(patt)[source]¶
Test whether the permutation
self
avoids the patternpatt
.EXAMPLES:
sage: Permutation([6,2,5,4,3,1]).avoids([4,2,3,1]) # needs sage.combinat False sage: Permutation([6,1,2,5,4,3]).avoids([4,2,3,1]) # needs sage.combinat True sage: Permutation([6,1,2,5,4,3]).avoids([3,4,1,2]) # needs sage.combinat True
>>> from sage.all import * >>> Permutation([Integer(6),Integer(2),Integer(5),Integer(4),Integer(3),Integer(1)]).avoids([Integer(4),Integer(2),Integer(3),Integer(1)]) # needs sage.combinat False >>> Permutation([Integer(6),Integer(1),Integer(2),Integer(5),Integer(4),Integer(3)]).avoids([Integer(4),Integer(2),Integer(3),Integer(1)]) # needs sage.combinat True >>> Permutation([Integer(6),Integer(1),Integer(2),Integer(5),Integer(4),Integer(3)]).avoids([Integer(3),Integer(4),Integer(1),Integer(2)]) # needs sage.combinat True
- binary_search_tree(left_to_right=True)[source]¶
Return the binary search tree associated to
self
.If \(w\) is a word, then the binary search tree associated to \(w\) is defined as the result of starting with an empty binary tree, and then inserting the letters of \(w\) one by one into this tree. Here, the insertion is being done according to the method
binary_search_insert()
, and the word \(w\) is being traversed from left to right.A permutation is regarded as a word (using one-line notation), and thus a binary search tree associated to a permutation is defined.
If the optional keyword variable
left_to_right
is set toFalse
, the word \(w\) is being traversed from right to left instead.EXAMPLES:
sage: Permutation([1,4,3,2]).binary_search_tree() # needs sage.graphs 1[., 4[3[2[., .], .], .]] sage: Permutation([4,1,3,2]).binary_search_tree() # needs sage.graphs 4[1[., 3[2[., .], .]], .]
>>> from sage.all import * >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).binary_search_tree() # needs sage.graphs 1[., 4[3[2[., .], .], .]] >>> Permutation([Integer(4),Integer(1),Integer(3),Integer(2)]).binary_search_tree() # needs sage.graphs 4[1[., 3[2[., .], .]], .]
By passing the option
left_to_right=False
one can have the insertion going from right to left:sage: Permutation([1,4,3,2]).binary_search_tree(False) # needs sage.graphs 2[1[., .], 3[., 4[., .]]] sage: Permutation([4,1,3,2]).binary_search_tree(False) # needs sage.graphs 2[1[., .], 3[., 4[., .]]]
>>> from sage.all import * >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).binary_search_tree(False) # needs sage.graphs 2[1[., .], 3[., 4[., .]]] >>> Permutation([Integer(4),Integer(1),Integer(3),Integer(2)]).binary_search_tree(False) # needs sage.graphs 2[1[., .], 3[., 4[., .]]]
- binary_search_tree_shape(left_to_right=True)[source]¶
Return the shape of the binary search tree of the permutation (a non labelled binary tree).
EXAMPLES:
sage: Permutation([1,4,3,2]).binary_search_tree_shape() # needs sage.graphs [., [[[., .], .], .]] sage: Permutation([4,1,3,2]).binary_search_tree_shape() # needs sage.graphs [[., [[., .], .]], .]
>>> from sage.all import * >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).binary_search_tree_shape() # needs sage.graphs [., [[[., .], .], .]] >>> Permutation([Integer(4),Integer(1),Integer(3),Integer(2)]).binary_search_tree_shape() # needs sage.graphs [[., [[., .], .]], .]
By passing the option
left_to_right=False
one can have the insertion going from right to left:sage: Permutation([1,4,3,2]).binary_search_tree_shape(False) # needs sage.graphs [[., .], [., [., .]]] sage: Permutation([4,1,3,2]).binary_search_tree_shape(False) # needs sage.graphs [[., .], [., [., .]]]
>>> from sage.all import * >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).binary_search_tree_shape(False) # needs sage.graphs [[., .], [., [., .]]] >>> Permutation([Integer(4),Integer(1),Integer(3),Integer(2)]).binary_search_tree_shape(False) # needs sage.graphs [[., .], [., [., .]]]
- bruhat_greater()[source]¶
Return the combinatorial class of permutations greater than or equal to
self
in the Bruhat order (on the symmetric group containingself
).See
bruhat_lequal()
for the definition of the Bruhat order.EXAMPLES:
sage: Permutation([4,1,2,3]).bruhat_greater().list() [[4, 1, 2, 3], [4, 1, 3, 2], [4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 1, 2], [4, 3, 2, 1]]
>>> from sage.all import * >>> Permutation([Integer(4),Integer(1),Integer(2),Integer(3)]).bruhat_greater().list() [[4, 1, 2, 3], [4, 1, 3, 2], [4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 1, 2], [4, 3, 2, 1]]
- bruhat_inversions()[source]¶
Return the list of inversions of
self
such that the application of this inversion toself
decreases its number of inversions by exactly 1.Equivalently, it returns the list of pairs \((i,j)\) such that \(i < j\), such that \(p(i) > p(j)\) and such that there exists no \(k\) (strictly) between \(i\) and \(j\) satisfying \(p(i) > p(k) > p(j)\).
EXAMPLES:
sage: Permutation([5,2,3,4,1]).bruhat_inversions() [[0, 1], [0, 2], [0, 3], [1, 4], [2, 4], [3, 4]] sage: Permutation([6,1,4,5,2,3]).bruhat_inversions() [[0, 1], [0, 2], [0, 3], [2, 4], [2, 5], [3, 4], [3, 5]]
>>> from sage.all import * >>> Permutation([Integer(5),Integer(2),Integer(3),Integer(4),Integer(1)]).bruhat_inversions() [[0, 1], [0, 2], [0, 3], [1, 4], [2, 4], [3, 4]] >>> Permutation([Integer(6),Integer(1),Integer(4),Integer(5),Integer(2),Integer(3)]).bruhat_inversions() [[0, 1], [0, 2], [0, 3], [2, 4], [2, 5], [3, 4], [3, 5]]
- bruhat_inversions_iterator()[source]¶
Return the iterator for the inversions of
self
such that the application of this inversion toself
decreases its number of inversions by exactly 1.EXAMPLES:
sage: list(Permutation([5,2,3,4,1]).bruhat_inversions_iterator()) [[0, 1], [0, 2], [0, 3], [1, 4], [2, 4], [3, 4]] sage: list(Permutation([6,1,4,5,2,3]).bruhat_inversions_iterator()) [[0, 1], [0, 2], [0, 3], [2, 4], [2, 5], [3, 4], [3, 5]]
>>> from sage.all import * >>> list(Permutation([Integer(5),Integer(2),Integer(3),Integer(4),Integer(1)]).bruhat_inversions_iterator()) [[0, 1], [0, 2], [0, 3], [1, 4], [2, 4], [3, 4]] >>> list(Permutation([Integer(6),Integer(1),Integer(4),Integer(5),Integer(2),Integer(3)]).bruhat_inversions_iterator()) [[0, 1], [0, 2], [0, 3], [2, 4], [2, 5], [3, 4], [3, 5]]
- bruhat_lequal(p2)[source]¶
Return
True
ifself
is less or equal top2
in the Bruhat order.The Bruhat order (also called strong Bruhat order or Chevalley order) on the symmetric group \(S_n\) is the partial order on \(S_n\) determined by the following condition: If \(p\) is a permutation, and \(i\) and \(j\) are two indices satisfying \(p(i) > p(j)\) and \(i < j\) (that is, \((i, j)\) is an inversion of \(p\) with \(i < j\)), then \(p \circ (i, j)\) (the permutation obtained by first switching \(i\) with \(j\) and then applying \(p\)) is smaller than \(p\) in the Bruhat order.
One can show that a permutation \(p \in S_n\) is less or equal to a permutation \(q \in S_n\) in the Bruhat order if and only if for every \(i \in \{ 0, 1, \cdots , n \}\) and \(j \in \{ 1, 2, \cdots , n \}\), the number of the elements among \(p(1), p(2), \cdots, p(j)\) that are greater than \(i\) is \(\leq\) to the number of the elements among \(q(1), q(2), \cdots, q(j)\) that are greater than \(i\).
This method assumes that
self
andp2
are permutations of the same integer \(n\).EXAMPLES:
sage: Permutation([2,4,3,1]).bruhat_lequal(Permutation([3,4,2,1])) True sage: Permutation([2,1,3]).bruhat_lequal(Permutation([2,3,1])) True sage: Permutation([2,1,3]).bruhat_lequal(Permutation([3,1,2])) True sage: Permutation([2,1,3]).bruhat_lequal(Permutation([1,2,3])) False sage: Permutation([1,3,2]).bruhat_lequal(Permutation([2,1,3])) False sage: Permutation([1,3,2]).bruhat_lequal(Permutation([2,3,1])) True sage: Permutation([2,3,1]).bruhat_lequal(Permutation([1,3,2])) False sage: sorted( [len([b for b in Permutations(3) if a.bruhat_lequal(b)]) ....: for a in Permutations(3)] ) [1, 2, 2, 4, 4, 6] sage: Permutation([]).bruhat_lequal(Permutation([])) True
>>> from sage.all import * >>> Permutation([Integer(2),Integer(4),Integer(3),Integer(1)]).bruhat_lequal(Permutation([Integer(3),Integer(4),Integer(2),Integer(1)])) True >>> Permutation([Integer(2),Integer(1),Integer(3)]).bruhat_lequal(Permutation([Integer(2),Integer(3),Integer(1)])) True >>> Permutation([Integer(2),Integer(1),Integer(3)]).bruhat_lequal(Permutation([Integer(3),Integer(1),Integer(2)])) True >>> Permutation([Integer(2),Integer(1),Integer(3)]).bruhat_lequal(Permutation([Integer(1),Integer(2),Integer(3)])) False >>> Permutation([Integer(1),Integer(3),Integer(2)]).bruhat_lequal(Permutation([Integer(2),Integer(1),Integer(3)])) False >>> Permutation([Integer(1),Integer(3),Integer(2)]).bruhat_lequal(Permutation([Integer(2),Integer(3),Integer(1)])) True >>> Permutation([Integer(2),Integer(3),Integer(1)]).bruhat_lequal(Permutation([Integer(1),Integer(3),Integer(2)])) False >>> sorted( [len([b for b in Permutations(Integer(3)) if a.bruhat_lequal(b)]) ... for a in Permutations(Integer(3))] ) [1, 2, 2, 4, 4, 6] >>> Permutation([]).bruhat_lequal(Permutation([])) True
- bruhat_pred()[source]¶
Return a list of the permutations strictly smaller than
self
in the Bruhat order (on the symmetric group containingself
) such that there is no permutation between one of those andself
.See
bruhat_lequal()
for the definition of the Bruhat order.EXAMPLES:
sage: Permutation([6,1,4,5,2,3]).bruhat_pred() [[1, 6, 4, 5, 2, 3], [4, 1, 6, 5, 2, 3], [5, 1, 4, 6, 2, 3], [6, 1, 2, 5, 4, 3], [6, 1, 3, 5, 2, 4], [6, 1, 4, 2, 5, 3], [6, 1, 4, 3, 2, 5]]
>>> from sage.all import * >>> Permutation([Integer(6),Integer(1),Integer(4),Integer(5),Integer(2),Integer(3)]).bruhat_pred() [[1, 6, 4, 5, 2, 3], [4, 1, 6, 5, 2, 3], [5, 1, 4, 6, 2, 3], [6, 1, 2, 5, 4, 3], [6, 1, 3, 5, 2, 4], [6, 1, 4, 2, 5, 3], [6, 1, 4, 3, 2, 5]]
- bruhat_pred_iterator()[source]¶
An iterator for the permutations strictly smaller than
self
in the Bruhat order (on the symmetric group containingself
) such that there is no permutation between one of those andself
.See
bruhat_lequal()
for the definition of the Bruhat order.EXAMPLES:
sage: [x for x in Permutation([6,1,4,5,2,3]).bruhat_pred_iterator()] [[1, 6, 4, 5, 2, 3], [4, 1, 6, 5, 2, 3], [5, 1, 4, 6, 2, 3], [6, 1, 2, 5, 4, 3], [6, 1, 3, 5, 2, 4], [6, 1, 4, 2, 5, 3], [6, 1, 4, 3, 2, 5]]
>>> from sage.all import * >>> [x for x in Permutation([Integer(6),Integer(1),Integer(4),Integer(5),Integer(2),Integer(3)]).bruhat_pred_iterator()] [[1, 6, 4, 5, 2, 3], [4, 1, 6, 5, 2, 3], [5, 1, 4, 6, 2, 3], [6, 1, 2, 5, 4, 3], [6, 1, 3, 5, 2, 4], [6, 1, 4, 2, 5, 3], [6, 1, 4, 3, 2, 5]]
- bruhat_smaller()[source]¶
Return the combinatorial class of permutations smaller than or equal to
self
in the Bruhat order (on the symmetric group containingself
).See
bruhat_lequal()
for the definition of the Bruhat order.EXAMPLES:
sage: Permutation([4,1,2,3]).bruhat_smaller().list() [[1, 2, 3, 4], [1, 2, 4, 3], [1, 3, 2, 4], [1, 4, 2, 3], [2, 1, 3, 4], [2, 1, 4, 3], [3, 1, 2, 4], [4, 1, 2, 3]]
>>> from sage.all import * >>> Permutation([Integer(4),Integer(1),Integer(2),Integer(3)]).bruhat_smaller().list() [[1, 2, 3, 4], [1, 2, 4, 3], [1, 3, 2, 4], [1, 4, 2, 3], [2, 1, 3, 4], [2, 1, 4, 3], [3, 1, 2, 4], [4, 1, 2, 3]]
- bruhat_succ()[source]¶
Return a list of the permutations strictly greater than
self
in the Bruhat order (on the symmetric group containingself
) such that there is no permutation between one of those andself
.See
bruhat_lequal()
for the definition of the Bruhat order.EXAMPLES:
sage: Permutation([6,1,4,5,2,3]).bruhat_succ() [[6, 4, 1, 5, 2, 3], [6, 2, 4, 5, 1, 3], [6, 1, 5, 4, 2, 3], [6, 1, 4, 5, 3, 2]]
>>> from sage.all import * >>> Permutation([Integer(6),Integer(1),Integer(4),Integer(5),Integer(2),Integer(3)]).bruhat_succ() [[6, 4, 1, 5, 2, 3], [6, 2, 4, 5, 1, 3], [6, 1, 5, 4, 2, 3], [6, 1, 4, 5, 3, 2]]
- bruhat_succ_iterator()[source]¶
An iterator for the permutations that are strictly greater than
self
in the Bruhat order (on the symmetric group containingself
) such that there is no permutation between one of those andself
.See
bruhat_lequal()
for the definition of the Bruhat order.EXAMPLES:
sage: [x for x in Permutation([6,1,4,5,2,3]).bruhat_succ_iterator()] [[6, 4, 1, 5, 2, 3], [6, 2, 4, 5, 1, 3], [6, 1, 5, 4, 2, 3], [6, 1, 4, 5, 3, 2]]
>>> from sage.all import * >>> [x for x in Permutation([Integer(6),Integer(1),Integer(4),Integer(5),Integer(2),Integer(3)]).bruhat_succ_iterator()] [[6, 4, 1, 5, 2, 3], [6, 2, 4, 5, 1, 3], [6, 1, 5, 4, 2, 3], [6, 1, 4, 5, 3, 2]]
- complement()[source]¶
Return the complement of the permutation
self
.The complement of a permutation \(w \in S_n\) is defined as the permutation in \(S_n\) sending each \(i\) to \(n + 1 - w(i)\).
EXAMPLES:
sage: Permutation([1,2,3]).complement() [3, 2, 1] sage: Permutation([1, 3, 2]).complement() [3, 1, 2]
>>> from sage.all import * >>> Permutation([Integer(1),Integer(2),Integer(3)]).complement() [3, 2, 1] >>> Permutation([Integer(1), Integer(3), Integer(2)]).complement() [3, 1, 2]
- cycle_string(singletons=False)[source]¶
Return a string of the permutation in cycle notation.
If
singletons=True
, it includes 1-cycles in the string.EXAMPLES:
sage: Permutation([1,2,3]).cycle_string() '()' sage: Permutation([2,1,3]).cycle_string() '(1,2)' sage: Permutation([2,3,1]).cycle_string() '(1,2,3)' sage: Permutation([2,1,3]).cycle_string(singletons=True) '(1,2)(3)'
>>> from sage.all import * >>> Permutation([Integer(1),Integer(2),Integer(3)]).cycle_string() '()' >>> Permutation([Integer(2),Integer(1),Integer(3)]).cycle_string() '(1,2)' >>> Permutation([Integer(2),Integer(3),Integer(1)]).cycle_string() '(1,2,3)' >>> Permutation([Integer(2),Integer(1),Integer(3)]).cycle_string(singletons=True) '(1,2)(3)'
- cycle_tuples(singletons=True, use_min=True)[source]¶
Return the permutation
self
as a list of disjoint cycles.The cycles are returned in the order of increasing smallest elements, and each cycle is returned as a tuple which starts with its smallest element.
If
singletons=False
is given, the list does not contain the singleton cycles.If
use_min=False
is given, the cycles are returned in the order of increasing largest (not smallest) elements, and each cycle starts with its largest element.EXAMPLES:
sage: Permutation([2,1,3,4]).to_cycles() [(1, 2), (3,), (4,)] sage: Permutation([2,1,3,4]).to_cycles(singletons=False) [(1, 2)] sage: Permutation([2,1,3,4]).to_cycles(use_min=True) [(1, 2), (3,), (4,)] sage: Permutation([2,1,3,4]).to_cycles(use_min=False) [(4,), (3,), (2, 1)] sage: Permutation([2,1,3,4]).to_cycles(singletons=False, use_min=False) [(2, 1)] sage: Permutation([4,1,5,2,6,3]).to_cycles() [(1, 4, 2), (3, 5, 6)] sage: Permutation([4,1,5,2,6,3]).to_cycles(use_min=False) [(6, 3, 5), (4, 2, 1)] sage: Permutation([6, 4, 5, 2, 3, 1]).to_cycles() [(1, 6), (2, 4), (3, 5)] sage: Permutation([6, 4, 5, 2, 3, 1]).to_cycles(use_min=False) [(6, 1), (5, 3), (4, 2)]
>>> from sage.all import * >>> Permutation([Integer(2),Integer(1),Integer(3),Integer(4)]).to_cycles() [(1, 2), (3,), (4,)] >>> Permutation([Integer(2),Integer(1),Integer(3),Integer(4)]).to_cycles(singletons=False) [(1, 2)] >>> Permutation([Integer(2),Integer(1),Integer(3),Integer(4)]).to_cycles(use_min=True) [(1, 2), (3,), (4,)] >>> Permutation([Integer(2),Integer(1),Integer(3),Integer(4)]).to_cycles(use_min=False) [(4,), (3,), (2, 1)] >>> Permutation([Integer(2),Integer(1),Integer(3),Integer(4)]).to_cycles(singletons=False, use_min=False) [(2, 1)] >>> Permutation([Integer(4),Integer(1),Integer(5),Integer(2),Integer(6),Integer(3)]).to_cycles() [(1, 4, 2), (3, 5, 6)] >>> Permutation([Integer(4),Integer(1),Integer(5),Integer(2),Integer(6),Integer(3)]).to_cycles(use_min=False) [(6, 3, 5), (4, 2, 1)] >>> Permutation([Integer(6), Integer(4), Integer(5), Integer(2), Integer(3), Integer(1)]).to_cycles() [(1, 6), (2, 4), (3, 5)] >>> Permutation([Integer(6), Integer(4), Integer(5), Integer(2), Integer(3), Integer(1)]).to_cycles(use_min=False) [(6, 1), (5, 3), (4, 2)]
The algorithm is of complexity \(O(n)\) where \(n\) is the size of the given permutation.
- cycle_type()[source]¶
Return a partition of
len(self)
corresponding to the cycle type ofself
.This is a non-increasing sequence of the cycle lengths of
self
.EXAMPLES:
sage: Permutation([3,1,2,4]).cycle_type() # needs sage.combinat [3, 1]
>>> from sage.all import * >>> Permutation([Integer(3),Integer(1),Integer(2),Integer(4)]).cycle_type() # needs sage.combinat [3, 1]
- decreasing_runs(as_tuple=False)[source]¶
Decreasing runs of the permutation.
INPUT:
as_tuple
– boolean (default:False
); choice of output format
OUTPUT: list of lists or a tuple of tuples
See also
EXAMPLES:
sage: s = Permutation([2,8,3,9,6,4,5,1,7]) sage: s.decreasing_runs() [[2], [8, 3], [9, 6, 4], [5, 1], [7]] sage: s.decreasing_runs(as_tuple=True) ((2,), (8, 3), (9, 6, 4), (5, 1), (7,))
>>> from sage.all import * >>> s = Permutation([Integer(2),Integer(8),Integer(3),Integer(9),Integer(6),Integer(4),Integer(5),Integer(1),Integer(7)]) >>> s.decreasing_runs() [[2], [8, 3], [9, 6, 4], [5, 1], [7]] >>> s.decreasing_runs(as_tuple=True) ((2,), (8, 3), (9, 6, 4), (5, 1), (7,))
- descent_polynomial()[source]¶
Return the descent polynomial of the permutation
self
.The descent polynomial of a permutation \(p\) is the product of all the
z[p(i)]
wherei
ranges over the descents ofp
.A descent of a permutation
p
is an integeri
such thatp(i) > p(i+1)
.REFERENCES:
EXAMPLES:
sage: Permutation([2,1,3]).descent_polynomial() z2 sage: Permutation([4,3,2,1]).descent_polynomial() z2*z3*z4
>>> from sage.all import * >>> Permutation([Integer(2),Integer(1),Integer(3)]).descent_polynomial() z2 >>> Permutation([Integer(4),Integer(3),Integer(2),Integer(1)]).descent_polynomial() z2*z3*z4
Todo
This docstring needs to be fixed. This is not defined in [GS1984] (the descent monomial in their (7.23) is different).
- descents(final_descent=False, side='right', positive=False, from_zero=False, index_set=None)[source]¶
Return the list of the descents of
self
.A descent of a permutation \(p\) is an integer \(i\) such that \(p(i) > p(i+1)\).
Warning
By default, the descents are returned as elements in the index set, i.e., starting at \(1\). If you want them to start at \(0\), set the keyword
from_zero
toTrue
.INPUT:
final_descent
– boolean (default:False
); ifTrue
, the last position of a non-empty permutation is also considered as a descentside
–'right'
(default) or'left'
; if'left'
, return the descents of the inverse permutationpositive
– boolean (default:False
); ifTrue
, return the positions that are not descentsfrom_zero
– boolean (default:False
); ifTrue
, return the positions starting from \(0\)index_set
– list (default:[1, ..., n-1]
whereself
is a permutation ofn
); the index set to check for descents
EXAMPLES:
sage: Permutation([3,1,2]).descents() [1] sage: Permutation([1,4,3,2]).descents() [2, 3] sage: Permutation([1,4,3,2]).descents(final_descent=True) [2, 3, 4] sage: Permutation([1,4,3,2]).descents(index_set=[1,2]) [2] sage: Permutation([1,4,3,2]).descents(from_zero=True) [1, 2]
>>> from sage.all import * >>> Permutation([Integer(3),Integer(1),Integer(2)]).descents() [1] >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).descents() [2, 3] >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).descents(final_descent=True) [2, 3, 4] >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).descents(index_set=[Integer(1),Integer(2)]) [2] >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).descents(from_zero=True) [1, 2]
- descents_composition()[source]¶
Return the descent composition of
self
.The descent composition of a permutation \(p \in S_n\) is defined as the composition of \(n\) whose descent set equals the descent set of \(p\). Here, the descent set of \(p\) is defined as the set of all \(i \in \{ 1, 2, \ldots, n-1 \}\) satisfying \(p(i) > p(i+1)\). The descent set of a composition \(c = (i_1, i_2, \ldots, i_k)\) is defined as the set \(\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}\).
EXAMPLES:
sage: Permutation([1,3,2,4]).descents_composition() [2, 2] sage: Permutation([4,1,6,7,2,3,8,5]).descents_composition() [1, 3, 3, 1] sage: Permutation([]).descents_composition() []
>>> from sage.all import * >>> Permutation([Integer(1),Integer(3),Integer(2),Integer(4)]).descents_composition() [2, 2] >>> Permutation([Integer(4),Integer(1),Integer(6),Integer(7),Integer(2),Integer(3),Integer(8),Integer(5)]).descents_composition() [1, 3, 3, 1] >>> Permutation([]).descents_composition() []
- destandardize(weight, ordered_alphabet=None)[source]¶
Return destandardization of
self
with respect toweight
andordered_alphabet
.INPUT:
weight
– list or tuple of nonnegative integers that sum to \(n\) ifself
is a permutation in \(S_n\)ordered_alphabet
– (default:None
) a list or tuple specifying the ordered alphabet the destandardized word is over
OUTPUT: word over the
ordered_alphabet
which standardizes toself
Let \(weight = (w_1,w_2,\ldots,w_\ell)\). Then this methods looks for an increasing sequence of \(1,2,\ldots, w_1\) and labels all letters in it by 1, then an increasing sequence of \(w_1+1,w_1+2,\ldots,w_1+w_2\) and labels all these letters by 2, etc.. If an increasing sequence for the specified
weight
does not exist, an error is returned. The output is a wordw
over the specified ordered alphabet with evaluationweight
such thatw.standard_permutation()
isself
.EXAMPLES:
sage: p = Permutation([1,2,5,3,6,4]) sage: p.destandardize([3,1,2]) # needs sage.combinat word: 113132 sage: p = Permutation([2,1,3]) sage: p.destandardize([2,1]) Traceback (most recent call last): ... ValueError: Standardization with weight [2, 1] is not possible!
>>> from sage.all import * >>> p = Permutation([Integer(1),Integer(2),Integer(5),Integer(3),Integer(6),Integer(4)]) >>> p.destandardize([Integer(3),Integer(1),Integer(2)]) # needs sage.combinat word: 113132 >>> p = Permutation([Integer(2),Integer(1),Integer(3)]) >>> p.destandardize([Integer(2),Integer(1)]) Traceback (most recent call last): ... ValueError: Standardization with weight [2, 1] is not possible!
- dict()[source]¶
Return a dictionary corresponding to the permutation.
EXAMPLES:
sage: p = Permutation([2,1,3]) sage: d = p.dict() sage: d[1] 2 sage: d[2] 1 sage: d[3] 3
>>> from sage.all import * >>> p = Permutation([Integer(2),Integer(1),Integer(3)]) >>> d = p.dict() >>> d[Integer(1)] 2 >>> d[Integer(2)] 1 >>> d[Integer(3)] 3
- fixed_points()[source]¶
Return a list of the fixed points of
self
.EXAMPLES:
sage: Permutation([1,3,2,4]).fixed_points() [1, 4] sage: Permutation([1,2,3,4]).fixed_points() [1, 2, 3, 4]
>>> from sage.all import * >>> Permutation([Integer(1),Integer(3),Integer(2),Integer(4)]).fixed_points() [1, 4] >>> Permutation([Integer(1),Integer(2),Integer(3),Integer(4)]).fixed_points() [1, 2, 3, 4]
- foata_bijection()[source]¶
Return the image of the permutation
self
under the Foata bijection \(\phi\).The bijection shows that \(\mathrm{maj}\) (the major index) and \(\mathrm{inv}\) (the number of inversions) are equidistributed: if \(\phi(P) = Q\), then \(\mathrm{maj}(P) = \mathrm{inv}(Q)\).
The Foata bijection \(\phi\) is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word: Given a word \(w_1 w_2 \cdots w_n\), start with \(\phi(w_1) = w_1\). At the \(i\)-th step, if \(\phi(w_1 w_2 \cdots w_i) = v_1 v_2 \cdots v_i\), we define \(\phi(w_1 w_2 \cdots w_i w_{i+1})\) by placing \(w_{i+1}\) on the end of the word \(v_1 v_2 \cdots v_i\) and breaking the word up into blocks as follows. If \(w_{i+1} > v_i\), place a vertical line to the right of each \(v_k\) for which \(w_{i+1} > v_k\). Otherwise, if \(w_{i+1} < v_i\), place a vertical line to the right of each \(v_k\) for which \(w_{i+1} < v_k\). In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
For instance, to compute \(\phi([1,4,2,5,3])\), the sequence of words is
\(1\),
\(|1|4 \to 14\),
\(|14|2 \to 412\),
\(|4|1|2|5 \to 4125\),
\(|4|125|3 \to 45123\).
So \(\phi([1,4,2,5,3]) = [4,5,1,2,3]\).
See section 2 of [FS1978], and the proof of Proposition 1.4.6 in [EnumComb1].
See also
foata_bijection_inverse()
for the inverse map.EXAMPLES:
sage: Permutation([1,2,4,3]).foata_bijection() [4, 1, 2, 3] sage: Permutation([2,5,1,3,4]).foata_bijection() [2, 1, 3, 5, 4] sage: P = Permutation([2,5,1,3,4]) sage: P.major_index() == P.foata_bijection().number_of_inversions() True sage: all( P.major_index() == P.foata_bijection().number_of_inversions() ....: for P in Permutations(4) ) True
>>> from sage.all import * >>> Permutation([Integer(1),Integer(2),Integer(4),Integer(3)]).foata_bijection() [4, 1, 2, 3] >>> Permutation([Integer(2),Integer(5),Integer(1),Integer(3),Integer(4)]).foata_bijection() [2, 1, 3, 5, 4] >>> P = Permutation([Integer(2),Integer(5),Integer(1),Integer(3),Integer(4)]) >>> P.major_index() == P.foata_bijection().number_of_inversions() True >>> all( P.major_index() == P.foata_bijection().number_of_inversions() ... for P in Permutations(Integer(4)) ) True
The example from [FS1978]:
sage: Permutation([7,4,9,2,6,1,5,8,3]).foata_bijection() [4, 7, 2, 6, 1, 9, 5, 8, 3]
>>> from sage.all import * >>> Permutation([Integer(7),Integer(4),Integer(9),Integer(2),Integer(6),Integer(1),Integer(5),Integer(8),Integer(3)]).foata_bijection() [4, 7, 2, 6, 1, 9, 5, 8, 3]
Border cases:
sage: Permutation([]).foata_bijection() [] sage: Permutation([1]).foata_bijection() [1]
>>> from sage.all import * >>> Permutation([]).foata_bijection() [] >>> Permutation([Integer(1)]).foata_bijection() [1]
- foata_bijection_inverse()[source]¶
Return the image of the permutation
self
under the inverse of the Foata bijection \(\phi\).See
foata_bijection()
for the definition of the Foata bijection.EXAMPLES:
sage: Permutation([4, 1, 2, 3]).foata_bijection() [1, 2, 4, 3]
>>> from sage.all import * >>> Permutation([Integer(4), Integer(1), Integer(2), Integer(3)]).foata_bijection() [1, 2, 4, 3]
- forget_cycles()[source]¶
Return the image of
self
under the map which forgets cycles.Consider a permutation \(\sigma\) written in standard cyclic form:
\[\sigma = (a_{1,1}, \ldots, a_{1,k_1}) (a_{2,1}, \ldots, a_{2,k_2}) \cdots (a_{m,1}, \ldots, a_{m,k_m}),\]where \(a_{1,1} < a_{2,1} < \cdots < a_{m,1}\) and \(a_{j,1} < a_{j,i}\) for all \(1 \leq j \leq m\) and \(2 \leq i \leq k_j\) where we include cycles of length 1 as well. The image of the forget cycle map \(\phi\) is given by
\[\phi(\sigma) = [a_{1,1}, \ldots, a_{1,k_1}, a_{2,1}, \ldots, a_{2,k_2}, \ldots, a_{m,1}, \ldots, a_{m,k_m}],\]considered as a permutation in 1-line notation.
See also
fundamental_transformation()
, which is a similar map that is defined by instead taking \(a_{j,1} > a_{j,i}\) and is bijective.EXAMPLES:
sage: P = Permutations(5) sage: x = P([1, 5, 3, 4, 2]) sage: x.forget_cycles() [1, 2, 5, 3, 4]
>>> from sage.all import * >>> P = Permutations(Integer(5)) >>> x = P([Integer(1), Integer(5), Integer(3), Integer(4), Integer(2)]) >>> x.forget_cycles() [1, 2, 5, 3, 4]
We select all permutations with a cycle composition of \([2, 3, 1]\) in \(S_6\):
sage: P = Permutations(6) sage: l = [p for p in P if [len(t) for t in p.to_cycles()] == [1,3,2]]
>>> from sage.all import * >>> P = Permutations(Integer(6)) >>> l = [p for p in P if [len(t) for t in p.to_cycles()] == [Integer(1),Integer(3),Integer(2)]]
Next we apply \(\phi\) and then take the inverse, and then view the results as a poset under the Bruhat order:
sage: l = [p.forget_cycles().inverse() for p in l] sage: B = Poset([l, lambda x,y: x.bruhat_lequal(y)]) # needs sage.combinat sage.graphs sage: R.<q> = QQ[] sage: sum(q^B.rank_function()(x) for x in B) # needs sage.combinat sage.graphs q^5 + 2*q^4 + 3*q^3 + 3*q^2 + 2*q + 1
>>> from sage.all import * >>> l = [p.forget_cycles().inverse() for p in l] >>> B = Poset([l, lambda x,y: x.bruhat_lequal(y)]) # needs sage.combinat sage.graphs >>> R = QQ['q']; (q,) = R._first_ngens(1) >>> sum(q**B.rank_function()(x) for x in B) # needs sage.combinat sage.graphs q^5 + 2*q^4 + 3*q^3 + 3*q^2 + 2*q + 1
We check the statement in [CC2013] that the posets \(C_{[1,3,1,1]}\) and \(C_{[1,3,2]}\) are isomorphic:
sage: l2 = [p for p in P if [len(t) for t in p.to_cycles()] == [1,3,1,1]] sage: l2 = [p.forget_cycles().inverse() for p in l2] sage: B2 = Poset([l2, lambda x,y: x.bruhat_lequal(y)]) # needs sage.combinat sage.graphs sage: B.is_isomorphic(B2) # needs sage.combinat sage.graphs True
>>> from sage.all import * >>> l2 = [p for p in P if [len(t) for t in p.to_cycles()] == [Integer(1),Integer(3),Integer(1),Integer(1)]] >>> l2 = [p.forget_cycles().inverse() for p in l2] >>> B2 = Poset([l2, lambda x,y: x.bruhat_lequal(y)]) # needs sage.combinat sage.graphs >>> B.is_isomorphic(B2) # needs sage.combinat sage.graphs True
See also
- fundamental_transformation()[source]¶
Return the image of the permutation
self
under the Renyi-Foata-Schuetzenberger fundamental transformation.The fundamental transformation is a bijection from the set of all permutations of \(\{1, 2, \ldots, n\}\) to itself, which transforms any such permutation \(w\) as follows: Write \(w\) in cycle form, with each cycle starting with its highest element, and the cycles being sorted in increasing order of their highest elements. Drop the parentheses in the resulting expression, thus reading it as a one-line notation of a new permutation \(u\). Then, \(u\) is the image of \(w\) under the fundamental transformation.
See [EnumComb1], Proposition 1.3.1.
See also
fundamental_transformation_inverse()
for the inverse map.forget_cycles()
for a similar (but non-bijective) map where each cycle is starting from its lowest element.EXAMPLES:
sage: Permutation([5, 1, 3, 4, 2]).fundamental_transformation() [3, 4, 5, 2, 1] sage: Permutations(5)([1, 5, 3, 4, 2]).fundamental_transformation() [1, 3, 4, 5, 2] sage: Permutation([8, 4, 7, 2, 9, 6, 5, 1, 3]).fundamental_transformation() [4, 2, 6, 8, 1, 9, 3, 7, 5]
>>> from sage.all import * >>> Permutation([Integer(5), Integer(1), Integer(3), Integer(4), Integer(2)]).fundamental_transformation() [3, 4, 5, 2, 1] >>> Permutations(Integer(5))([Integer(1), Integer(5), Integer(3), Integer(4), Integer(2)]).fundamental_transformation() [1, 3, 4, 5, 2] >>> Permutation([Integer(8), Integer(4), Integer(7), Integer(2), Integer(9), Integer(6), Integer(5), Integer(1), Integer(3)]).fundamental_transformation() [4, 2, 6, 8, 1, 9, 3, 7, 5]
Comparison with
forget_cycles()
:sage: P = Permutation([(1,3,4),(2,5)]) sage: P [3, 5, 4, 1, 2] sage: P.forget_cycles() [1, 3, 4, 2, 5] sage: P.fundamental_transformation() [4, 1, 3, 5, 2]
>>> from sage.all import * >>> P = Permutation([(Integer(1),Integer(3),Integer(4)),(Integer(2),Integer(5))]) >>> P [3, 5, 4, 1, 2] >>> P.forget_cycles() [1, 3, 4, 2, 5] >>> P.fundamental_transformation() [4, 1, 3, 5, 2]
- fundamental_transformation_inverse()[source]¶
Return the image of the permutation
self
under the inverse of the Renyi-Foata-Schuetzenberger fundamental transformation.The inverse of the fundamental transformation is a bijection from the set of all permutations of \(\{1, 2, \ldots, n\}\) to itself, which transforms any such permutation \(w\) as follows: Let \(I = \{ i_1 < i_2 < \cdots < i_k \}\) be the set of all left-to-right maxima of \(w\) (that is, of all indices \(j\) such that \(w(j)\) is bigger than each of \(w(1), w(2), \ldots, w(j-1)\)). The image of \(w\) under the inverse of the fundamental transformation is the permutation \(u\) that sends \(w(i-1)\) to \(w(i)\) for all \(i \notin I\) (notice that this makes sense, since \(1 \in I\) whenever \(n > 0\)), while sending each \(w(i_p - 1)\) (with \(p \geq 2\)) to \(w(i_{p-1})\). Here, we set \(i_{k+1} = n+1\).
See [EnumComb1], Proposition 1.3.1.
See also
fundamental_transformation()
for the inverse map.EXAMPLES:
sage: Permutation([3, 4, 5, 2, 1]).fundamental_transformation_inverse() [5, 1, 3, 4, 2] sage: Permutation([4, 2, 6, 8, 1, 9, 3, 7, 5]).fundamental_transformation_inverse() [8, 4, 7, 2, 9, 6, 5, 1, 3]
>>> from sage.all import * >>> Permutation([Integer(3), Integer(4), Integer(5), Integer(2), Integer(1)]).fundamental_transformation_inverse() [5, 1, 3, 4, 2] >>> Permutation([Integer(4), Integer(2), Integer(6), Integer(8), Integer(1), Integer(9), Integer(3), Integer(7), Integer(5)]).fundamental_transformation_inverse() [8, 4, 7, 2, 9, 6, 5, 1, 3]
- grade()[source]¶
Return the size of
self
.EXAMPLES:
sage: Permutation([3,4,1,2,5]).size() 5
>>> from sage.all import * >>> Permutation([Integer(3),Integer(4),Integer(1),Integer(2),Integer(5)]).size() 5
- has_nth_root(n)[source]¶
Check if
self
has \(n\)-th roots.An \(n\)-th root of the permutation \(\sigma\) is a permutation \(\gamma\) such that \(\gamma^n = \sigma\).
Note that the number of \(n\)-th roots only depends on the cycle type of
self
.EXAMPLES:
sage: # needs sage.combinat sage: sigma = Permutations(5).identity() sage: sigma.has_nth_root(3) True sage: sigma = Permutation('(1, 3)') sage: sigma.has_nth_root(2) False
>>> from sage.all import * >>> # needs sage.combinat >>> sigma = Permutations(Integer(5)).identity() >>> sigma.has_nth_root(Integer(3)) True >>> sigma = Permutation('(1, 3)') >>> sigma.has_nth_root(Integer(2)) False
See also
- has_pattern(patt)[source]¶
Test whether the permutation
self
contains the patternpatt
.EXAMPLES:
sage: Permutation([3,5,1,4,6,2]).has_pattern([1,3,2]) # needs sage.combinat True
>>> from sage.all import * >>> Permutation([Integer(3),Integer(5),Integer(1),Integer(4),Integer(6),Integer(2)]).has_pattern([Integer(1),Integer(3),Integer(2)]) # needs sage.combinat True
- hyperoctahedral_double_coset_type()[source]¶
Return the coset-type of
self
as a partition.self
must be a permutation of even size \(2n\). The coset-type determines the double class of the permutation, that is its image in \(H_n \backslash S_{2n} / H_n\), where \(H_n\) is the \(n\)-th hyperoctahedral group.The coset-type is determined as follows. Consider the perfect matching \(\{\{1,2\},\{3,4\},\dots,\{2n-1,2n\}\}\) and its image by
self
, and draw them simultaneously as edges of a graph whose vertices are labeled by \(1,2,\dots,2n\). The coset-type is the ordered sequence of the semi-lengths of the cycles of this graph (see Chapter VII of [Mac1995] for more details, particularly Section VII.2).EXAMPLES:
sage: # needs sage.combinat sage: p = Permutation([3, 4, 6, 1, 5, 7, 2, 8]) sage: p.hyperoctahedral_double_coset_type() [3, 1] sage: all(p.hyperoctahedral_double_coset_type() == ....: p.inverse().hyperoctahedral_double_coset_type() ....: for p in Permutations(4)) True sage: Permutation([]).hyperoctahedral_double_coset_type() [] sage: Permutation([3,1,2]).hyperoctahedral_double_coset_type() Traceback (most recent call last): ... ValueError: [3, 1, 2] is a permutation of odd size and has no coset-type
>>> from sage.all import * >>> # needs sage.combinat >>> p = Permutation([Integer(3), Integer(4), Integer(6), Integer(1), Integer(5), Integer(7), Integer(2), Integer(8)]) >>> p.hyperoctahedral_double_coset_type() [3, 1] >>> all(p.hyperoctahedral_double_coset_type() == ... p.inverse().hyperoctahedral_double_coset_type() ... for p in Permutations(Integer(4))) True >>> Permutation([]).hyperoctahedral_double_coset_type() [] >>> Permutation([Integer(3),Integer(1),Integer(2)]).hyperoctahedral_double_coset_type() Traceback (most recent call last): ... ValueError: [3, 1, 2] is a permutation of odd size and has no coset-type
- idescents(final_descent=False, from_zero=False)[source]¶
Return a list of the idescents of
self
, that is the list of the descents ofself
’s inverse.A descent of a permutation
p
is an integeri
such thatp(i) > p(i+1)
.Warning
By default, the descents are returned as elements in the index set, i.e., starting at \(1\). If you want them to start at \(0\), set the keyword
from_zero
toTrue
.INPUT:
final_descent
– boolean (default:False
); ifTrue
, the last position of a non-empty permutation is also considered as a descentfrom_zero
– boolean (default:False
); ifFalse
, return the positions starting from \(1\)
EXAMPLES:
sage: Permutation([2,3,1]).idescents() [1] sage: Permutation([1,4,3,2]).idescents() [2, 3] sage: Permutation([1,4,3,2]).idescents(final_descent=True) [2, 3, 4] sage: Permutation([1,4,3,2]).idescents(from_zero=True) [1, 2]
>>> from sage.all import * >>> Permutation([Integer(2),Integer(3),Integer(1)]).idescents() [1] >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).idescents() [2, 3] >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).idescents(final_descent=True) [2, 3, 4] >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).idescents(from_zero=True) [1, 2]
- idescents_signature(final_descent=False)[source]¶
Return the list obtained as follows: Each position in
self
is mapped to \(-1\) if it is an idescent and \(1\) if it is not an idescent.See
idescents()
for a definition of idescents.With the
final_descent
option, the last position of a non-empty permutation is also considered as a descent.EXAMPLES:
sage: Permutation([1,4,3,2]).idescents() [2, 3] sage: Permutation([1,4,3,2]).idescents_signature() [1, -1, -1, 1]
>>> from sage.all import * >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).idescents() [2, 3] >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).idescents_signature() [1, -1, -1, 1]
- imajor_index(final_descent=False)[source]¶
Return the inverse major index of the permutation
self
, which is the major index of the inverse ofself
.The major index of a permutation \(p\) is the sum of the descents of \(p\). Since our permutation indices are 0-based, we need to add the number of descents.
With the
final_descent
option, the last position of a non-empty permutation is also considered as a descent.EXAMPLES:
sage: Permutation([2,1,3]).imajor_index() 1 sage: Permutation([3,4,1,2]).imajor_index() 2 sage: Permutation([4,3,2,1]).imajor_index() 6
>>> from sage.all import * >>> Permutation([Integer(2),Integer(1),Integer(3)]).imajor_index() 1 >>> Permutation([Integer(3),Integer(4),Integer(1),Integer(2)]).imajor_index() 2 >>> Permutation([Integer(4),Integer(3),Integer(2),Integer(1)]).imajor_index() 6
- increasing_tree(compare=<built-in function min>)[source]¶
Return the increasing tree associated to
self
.EXAMPLES:
sage: Permutation([1,4,3,2]).increasing_tree() # needs sage.graphs 1[., 2[3[4[., .], .], .]] sage: Permutation([4,1,3,2]).increasing_tree() # needs sage.graphs 1[4[., .], 2[3[., .], .]]
>>> from sage.all import * >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).increasing_tree() # needs sage.graphs 1[., 2[3[4[., .], .], .]] >>> Permutation([Integer(4),Integer(1),Integer(3),Integer(2)]).increasing_tree() # needs sage.graphs 1[4[., .], 2[3[., .], .]]
By passing the option
compare=max
one can have the decreasing tree instead:sage: Permutation([2,3,4,1]).increasing_tree(max) # needs sage.graphs 4[3[2[., .], .], 1[., .]] sage: Permutation([2,3,1,4]).increasing_tree(max) # needs sage.graphs 4[3[2[., .], 1[., .]], .]
>>> from sage.all import * >>> Permutation([Integer(2),Integer(3),Integer(4),Integer(1)]).increasing_tree(max) # needs sage.graphs 4[3[2[., .], .], 1[., .]] >>> Permutation([Integer(2),Integer(3),Integer(1),Integer(4)]).increasing_tree(max) # needs sage.graphs 4[3[2[., .], 1[., .]], .]
- increasing_tree_shape(compare=<built-in function min>)[source]¶
Return the shape of the increasing tree associated with the permutation.
EXAMPLES:
sage: Permutation([1,4,3,2]).increasing_tree_shape() # needs sage.graphs [., [[[., .], .], .]] sage: Permutation([4,1,3,2]).increasing_tree_shape() # needs sage.graphs [[., .], [[., .], .]]
>>> from sage.all import * >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).increasing_tree_shape() # needs sage.graphs [., [[[., .], .], .]] >>> Permutation([Integer(4),Integer(1),Integer(3),Integer(2)]).increasing_tree_shape() # needs sage.graphs [[., .], [[., .], .]]
By passing the option
compare=max
one can have the decreasing tree instead:sage: Permutation([2,3,4,1]).increasing_tree_shape(max) # needs sage.graphs [[[., .], .], [., .]] sage: Permutation([2,3,1,4]).increasing_tree_shape(max) # needs sage.graphs [[[., .], [., .]], .]
>>> from sage.all import * >>> Permutation([Integer(2),Integer(3),Integer(4),Integer(1)]).increasing_tree_shape(max) # needs sage.graphs [[[., .], .], [., .]] >>> Permutation([Integer(2),Integer(3),Integer(1),Integer(4)]).increasing_tree_shape(max) # needs sage.graphs [[[., .], [., .]], .]
- inverse()[source]¶
Return the inverse of
self
.EXAMPLES:
sage: Permutation([3,8,5,10,9,4,6,1,7,2]).inverse() [8, 10, 1, 6, 3, 7, 9, 2, 5, 4] sage: Permutation([2, 4, 1, 5, 3]).inverse() [3, 1, 5, 2, 4] sage: ~Permutation([2, 4, 1, 5, 3]) [3, 1, 5, 2, 4]
>>> from sage.all import * >>> Permutation([Integer(3),Integer(8),Integer(5),Integer(10),Integer(9),Integer(4),Integer(6),Integer(1),Integer(7),Integer(2)]).inverse() [8, 10, 1, 6, 3, 7, 9, 2, 5, 4] >>> Permutation([Integer(2), Integer(4), Integer(1), Integer(5), Integer(3)]).inverse() [3, 1, 5, 2, 4] >>> ~Permutation([Integer(2), Integer(4), Integer(1), Integer(5), Integer(3)]) [3, 1, 5, 2, 4]
- inversions()[source]¶
Return a list of the inversions of
self
.An inversion of a permutation \(p\) is a pair \((i, j)\) such that \(i < j\) and \(p(i) > p(j)\).
EXAMPLES:
sage: Permutation([3,2,4,1,5]).inversions() [(1, 2), (1, 4), (2, 4), (3, 4)]
>>> from sage.all import * >>> Permutation([Integer(3),Integer(2),Integer(4),Integer(1),Integer(5)]).inversions() [(1, 2), (1, 4), (2, 4), (3, 4)]
- is_derangement()[source]¶
Return whether
self
is a derangement.A permutation \(\sigma\) is a derangement if \(\sigma\) has no fixed points.
EXAMPLES:
sage: P = Permutation([1,4,2,3]) sage: P.is_derangement() False sage: P = Permutation([2,3,1]) sage: P.is_derangement() True
>>> from sage.all import * >>> P = Permutation([Integer(1),Integer(4),Integer(2),Integer(3)]) >>> P.is_derangement() False >>> P = Permutation([Integer(2),Integer(3),Integer(1)]) >>> P.is_derangement() True
- is_even()[source]¶
Return
True
if the permutationself
is even andFalse
otherwise.EXAMPLES:
sage: Permutation([1,2,3]).is_even() True sage: Permutation([2,1,3]).is_even() False
>>> from sage.all import * >>> Permutation([Integer(1),Integer(2),Integer(3)]).is_even() True >>> Permutation([Integer(2),Integer(1),Integer(3)]).is_even() False
- is_simple()[source]¶
Return whether
self
is simple.A permutation is simple if it does not send any proper sub-interval to a sub-interval.
For instance,
[6,1,3,5,2,4]
is not simple because it maps the interval[3,4,5,6]
to[2,3,4,5]
, whereas[2,6,3,5,1,4]
is simple.EXAMPLES:
sage: g = Permutation([4,2,3,1]) sage: g.is_simple() False sage: g = Permutation([6,1,3,5,2,4]) sage: g.is_simple() False sage: g = Permutation([2,6,3,5,1,4]) sage: g.is_simple() True sage: [len([pi for pi in Permutations(n) if pi.is_simple()]) ....: for n in range(6)] [1, 1, 2, 0, 2, 6]
>>> from sage.all import * >>> g = Permutation([Integer(4),Integer(2),Integer(3),Integer(1)]) >>> g.is_simple() False >>> g = Permutation([Integer(6),Integer(1),Integer(3),Integer(5),Integer(2),Integer(4)]) >>> g.is_simple() False >>> g = Permutation([Integer(2),Integer(6),Integer(3),Integer(5),Integer(1),Integer(4)]) >>> g.is_simple() True >>> [len([pi for pi in Permutations(n) if pi.is_simple()]) ... for n in range(Integer(6))] [1, 1, 2, 0, 2, 6]
- ishift(i)[source]¶
Return the
i
-shift ofself
. If ani
-shift ofself
can’t be performed, thenself
is returned.An \(i\)-shift can be applied when \(i\) is not inbetween \(i-1\) and \(i+1\). The \(i\)-shift moves \(i\) to the other side, and leaves the relative positions of \(i-1\) and \(i+1\) in place. All other entries of the permutations are also left in place.
EXAMPLES:
Here, \(2\) is to the left of both \(1\) and \(3\). A \(2\)-shift can be applied which moves the \(2\) to the right and leaves \(1\) and \(3\) in their same relative order:
sage: Permutation([2,1,3]).ishift(2) [1, 3, 2]
>>> from sage.all import * >>> Permutation([Integer(2),Integer(1),Integer(3)]).ishift(Integer(2)) [1, 3, 2]
All entries other than \(i\), \(i-1\) and \(i+1\) are unchanged:
sage: Permutation([2,4,1,3]).ishift(2) [1, 4, 3, 2]
>>> from sage.all import * >>> Permutation([Integer(2),Integer(4),Integer(1),Integer(3)]).ishift(Integer(2)) [1, 4, 3, 2]
Since \(2\) is between \(1\) and \(3\) in
[1,2,3]
, a \(2\)-shift cannot be applied to[1,2,3]
sage: Permutation([1,2,3]).ishift(2) [1, 2, 3]
>>> from sage.all import * >>> Permutation([Integer(1),Integer(2),Integer(3)]).ishift(Integer(2)) [1, 2, 3]
- iswitch(i)[source]¶
Return the
i
-switch ofself
. If ani
-switch ofself
can’t be performed, thenself
is returned.An \(i\)-switch can be applied when the subsequence of
self
formed by the entries \(i-1\), \(i\) and \(i+1\) is neither increasing nor decreasing. In this case, this subsequence is reversed (i. e., its leftmost element and its rightmost element switch places), while all other letters ofself
are kept in place.EXAMPLES:
Here, \(2\) is to the left of both \(1\) and \(3\). A \(2\)-switch can be applied which moves the \(2\) to the right and switches the relative order between \(1\) and \(3\):
sage: Permutation([2,1,3]).iswitch(2) [3, 1, 2]
>>> from sage.all import * >>> Permutation([Integer(2),Integer(1),Integer(3)]).iswitch(Integer(2)) [3, 1, 2]
All entries other than \(i-1\), \(i\) and \(i+1\) are unchanged:
sage: Permutation([2,4,1,3]).iswitch(2) [3, 4, 1, 2]
>>> from sage.all import * >>> Permutation([Integer(2),Integer(4),Integer(1),Integer(3)]).iswitch(Integer(2)) [3, 4, 1, 2]
Since \(2\) is between \(1\) and \(3\) in
[1,2,3]
, a \(2\)-switch cannot be applied to[1,2,3]
sage: Permutation([1,2,3]).iswitch(2) [1, 2, 3]
>>> from sage.all import * >>> Permutation([Integer(1),Integer(2),Integer(3)]).iswitch(Integer(2)) [1, 2, 3]
- left_action_product(lp)[source]¶
Return the permutation obtained by composing
self
withlp
in such an order thatlp
is applied first andself
is applied afterwards.This is usually denoted by either
self * lp
orlp * self
depending on the conventions used by the author. If the value of a permutation \(p \in S_n\) on an integer \(i \in \{ 1, 2, \cdots, n \}\) is denoted by \(p(i)\), then this should be denoted byself * lp
in order to have associativity (i.e., in order to have \((p \cdot q)(i) = p(q(i))\) for all \(p\), \(q\) and \(i\)). If, on the other hand, the value of a permutation \(p \in S_n\) on an integer \(i \in \{ 1, 2, \cdots, n \}\) is denoted by \(i^p\), then this should be denoted bylp * self
in order to have associativity (i.e., in order to have \(i^{p \cdot q} = (i^p)^q\) for all \(p\), \(q\) and \(i\)).EXAMPLES:
sage: p = Permutation([2,1,3]) sage: q = Permutation([3,1,2]) sage: p.left_action_product(q) [3, 2, 1] sage: q.left_action_product(p) [1, 3, 2]
>>> from sage.all import * >>> p = Permutation([Integer(2),Integer(1),Integer(3)]) >>> q = Permutation([Integer(3),Integer(1),Integer(2)]) >>> p.left_action_product(q) [3, 2, 1] >>> q.left_action_product(p) [1, 3, 2]
- left_tableau()[source]¶
Return the left standard tableau after performing the RSK algorithm on
self
.EXAMPLES:
sage: Permutation([1,4,3,2]).left_tableau() # needs sage.combinat [[1, 2], [3], [4]]
>>> from sage.all import * >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).left_tableau() # needs sage.combinat [[1, 2], [3], [4]]
- length()[source]¶
Return the Coxeter length of
self
.The length of a permutation \(p\) is given by the number of inversions of \(p\).
EXAMPLES:
sage: Permutation([5, 1, 3, 4, 2]).length() 6
>>> from sage.all import * >>> Permutation([Integer(5), Integer(1), Integer(3), Integer(4), Integer(2)]).length() 6
- longest_increasing_subsequence_length()[source]¶
Return the length of the longest increasing subsequences of
self
.EXAMPLES:
sage: Permutation([2,3,1,4]).longest_increasing_subsequence_length() 3 sage: all(i.longest_increasing_subsequence_length() == len(RSK(i)[0][0]) # needs sage.combinat ....: for i in Permutations(5)) True sage: Permutation([]).longest_increasing_subsequence_length() 0
>>> from sage.all import * >>> Permutation([Integer(2),Integer(3),Integer(1),Integer(4)]).longest_increasing_subsequence_length() 3 >>> all(i.longest_increasing_subsequence_length() == len(RSK(i)[Integer(0)][Integer(0)]) # needs sage.combinat ... for i in Permutations(Integer(5))) True >>> Permutation([]).longest_increasing_subsequence_length() 0
- longest_increasing_subsequences()[source]¶
Return the list of the longest increasing subsequences of
self
.A theorem of Schensted ([Sch1961]) states that an increasing subsequence of length \(i\) ends with the value entered in the \(i\)-th column of the p-tableau. The algorithm records which column of the p-tableau each value of the permutation is entered into, creates a digraph to record all increasing subsequences, and reads the paths from a source to a sink; these are the longest increasing subsequences.
EXAMPLES:
sage: Permutation([2,3,4,1]).longest_increasing_subsequences() # needs sage.graphs [[2, 3, 4]] sage: Permutation([5, 7, 1, 2, 6, 4, 3]).longest_increasing_subsequences() # needs sage.graphs [[1, 2, 6], [1, 2, 4], [1, 2, 3]]
>>> from sage.all import * >>> Permutation([Integer(2),Integer(3),Integer(4),Integer(1)]).longest_increasing_subsequences() # needs sage.graphs [[2, 3, 4]] >>> Permutation([Integer(5), Integer(7), Integer(1), Integer(2), Integer(6), Integer(4), Integer(3)]).longest_increasing_subsequences() # needs sage.graphs [[1, 2, 6], [1, 2, 4], [1, 2, 3]]
Note
This algorithm could be made faster using a balanced search tree for each column instead of sorted lists. See discussion on Issue #31451.
- longest_increasing_subsequences_number()[source]¶
Return the number of increasing subsequences of maximal length in
self
.The list of longest increasing subsequences of a permutation is given by
longest_increasing_subsequences()
, and the length of these subsequences is given bylongest_increasing_subsequence_length()
.The algorithm is similar to
longest_increasing_subsequences()
. Namely, the longest increasing subsequences are encoded as increasing sequences in a ranked poset from a smallest to a largest element. Their number can be obtained via dynamic programming: for each \(v\) in the poset we compute the number of paths from a smallest element to \(v\).EXAMPLES:
sage: sum(p.longest_increasing_subsequences_number() ....: for p in Permutations(8)) 120770 sage: p = Permutations(50).random_element() sage: (len(p.longest_increasing_subsequences()) == # needs sage.graphs ....: p.longest_increasing_subsequences_number()) True
>>> from sage.all import * >>> sum(p.longest_increasing_subsequences_number() ... for p in Permutations(Integer(8))) 120770 >>> p = Permutations(Integer(50)).random_element() >>> (len(p.longest_increasing_subsequences()) == # needs sage.graphs ... p.longest_increasing_subsequences_number()) True
- major_index(final_descent=False)[source]¶
Return the major index of
self
.The major index of a permutation \(p\) is the sum of the descents of \(p\). Since our permutation indices are 0-based, we need to add the number of descents.
With the
final_descent
option, the last position of a non-empty permutation is also considered as a descent.EXAMPLES:
sage: Permutation([2,1,3]).major_index() 1 sage: Permutation([3,4,1,2]).major_index() 2 sage: Permutation([4,3,2,1]).major_index() 6
>>> from sage.all import * >>> Permutation([Integer(2),Integer(1),Integer(3)]).major_index() 1 >>> Permutation([Integer(3),Integer(4),Integer(1),Integer(2)]).major_index() 2 >>> Permutation([Integer(4),Integer(3),Integer(2),Integer(1)]).major_index() 6
- multi_major_index(composition)[source]¶
Return the multimajor index of this permutation with respect to
composition
.INPUT:
composition
– a composition of thesize()
of this permutation
EXAMPLES:
sage: p = Permutation([5, 6, 2, 1, 3, 7, 4]) sage: p.multi_major_index([3, 2, 2]) [2, 0, 1] sage: p.multi_major_index([7]) == [p.major_index()] True sage: p.multi_major_index([1]*7) [0, 0, 0, 0, 0, 0, 0] sage: Permutation([]).multi_major_index([]) []
>>> from sage.all import * >>> p = Permutation([Integer(5), Integer(6), Integer(2), Integer(1), Integer(3), Integer(7), Integer(4)]) >>> p.multi_major_index([Integer(3), Integer(2), Integer(2)]) [2, 0, 1] >>> p.multi_major_index([Integer(7)]) == [p.major_index()] True >>> p.multi_major_index([Integer(1)]*Integer(7)) [0, 0, 0, 0, 0, 0, 0] >>> Permutation([]).multi_major_index([]) []
REFERENCES:
- next()[source]¶
Return the permutation that follows
self
on the symmetric group containingself
. Ifself
is the last permutation, thennext
returnsFalse
. If thealgorithm
parameter is specified, the permutations will be generated according to it. Supported algorithms are:lex
: lexicographic order generation, this is the default algorithm.sjt
: Steinhaus-Johnson-Trotter algorithm to generate permutations using only transposition of two elements in the list. It is highly recommended to setcheck=True
(default value).
EXAMPLES:
sage: p = Permutation([1, 3, 2]) sage: next(p) [2, 1, 3] sage: p = Permutation([4,3,2,1]) sage: next(p) False sage: p = Permutation([1, 2, 3], algorithm='sjt') sage: p = next(p); p [1, 3, 2] sage: p = next(p); p [3, 1, 2]
>>> from sage.all import * >>> p = Permutation([Integer(1), Integer(3), Integer(2)]) >>> next(p) [2, 1, 3] >>> p = Permutation([Integer(4),Integer(3),Integer(2),Integer(1)]) >>> next(p) False >>> p = Permutation([Integer(1), Integer(2), Integer(3)], algorithm='sjt') >>> p = next(p); p [1, 3, 2] >>> p = next(p); p [3, 1, 2]
- noninversions(k)[source]¶
Return the list of all
k
-noninversions inself
.If \(k\) is an integer and \(p \in S_n\) is a permutation, then a \(k\)-noninversion in \(p\) is defined as a strictly increasing sequence \((i_1, i_2, \ldots, i_k)\) of elements of \(\{ 1, 2, \ldots, n \}\) satisfying \(p(i_1) < p(i_2) < \cdots < p(i_k)\). (In other words, a \(k\)-noninversion in \(p\) can be regarded as a \(k\)-element subset of \(\{ 1, 2, \ldots, n \}\) on which \(p\) restricts to an increasing map.)
EXAMPLES:
sage: p = Permutation([3, 2, 4, 1, 5]) sage: p.noninversions(1) [[3], [2], [4], [1], [5]] sage: p.noninversions(2) [[3, 4], [3, 5], [2, 4], [2, 5], [4, 5], [1, 5]] sage: p.noninversions(3) [[3, 4, 5], [2, 4, 5]] sage: p.noninversions(4) [] sage: p.noninversions(5) []
>>> from sage.all import * >>> p = Permutation([Integer(3), Integer(2), Integer(4), Integer(1), Integer(5)]) >>> p.noninversions(Integer(1)) [[3], [2], [4], [1], [5]] >>> p.noninversions(Integer(2)) [[3, 4], [3, 5], [2, 4], [2, 5], [4, 5], [1, 5]] >>> p.noninversions(Integer(3)) [[3, 4, 5], [2, 4, 5]] >>> p.noninversions(Integer(4)) [] >>> p.noninversions(Integer(5)) []
- nth_roots(n)[source]¶
Return all \(n\)-th roots of
self
(as a generator).An \(n\)-th root of the permutation \(\sigma\) is a permutation \(\gamma\) such that \(\gamma^n = \sigma\).
Note that the number of \(n\)-th roots only depends on the cycle type of
self
.EXAMPLES:
sage: # needs sage.combinat sage: sigma = Permutations(5).identity() sage: list(sigma.nth_roots(3)) [[1, 4, 3, 5, 2], [1, 5, 3, 2, 4], [1, 2, 4, 5, 3], [1, 2, 5, 3, 4], [4, 2, 3, 5, 1], [5, 2, 3, 1, 4], [3, 2, 5, 4, 1], [5, 2, 1, 4, 3], [2, 5, 3, 4, 1], [5, 1, 3, 4, 2], [2, 3, 1, 4, 5], [3, 1, 2, 4, 5], [2, 4, 3, 1, 5], [4, 1, 3, 2, 5], [3, 2, 4, 1, 5], [4, 2, 1, 3, 5], [1, 3, 4, 2, 5], [1, 4, 2, 3, 5], [1, 3, 5, 4, 2], [1, 5, 2, 4, 3], [1, 2, 3, 4, 5]] sage: sigma = Permutation('(1, 3)') sage: list(sigma.nth_roots(2)) []
>>> from sage.all import * >>> # needs sage.combinat >>> sigma = Permutations(Integer(5)).identity() >>> list(sigma.nth_roots(Integer(3))) [[1, 4, 3, 5, 2], [1, 5, 3, 2, 4], [1, 2, 4, 5, 3], [1, 2, 5, 3, 4], [4, 2, 3, 5, 1], [5, 2, 3, 1, 4], [3, 2, 5, 4, 1], [5, 2, 1, 4, 3], [2, 5, 3, 4, 1], [5, 1, 3, 4, 2], [2, 3, 1, 4, 5], [3, 1, 2, 4, 5], [2, 4, 3, 1, 5], [4, 1, 3, 2, 5], [3, 2, 4, 1, 5], [4, 2, 1, 3, 5], [1, 3, 4, 2, 5], [1, 4, 2, 3, 5], [1, 3, 5, 4, 2], [1, 5, 2, 4, 3], [1, 2, 3, 4, 5]] >>> sigma = Permutation('(1, 3)') >>> list(sigma.nth_roots(Integer(2))) []
For \(n \geq 6\), this algorithm begins to be more efficient than naive search (look at all permutations and test their \(n\)-th power).
See also
- number_of_descents(final_descent=False)[source]¶
Return the number of descents of
self
.With the
final_descent
option, the last position of a non-empty permutation is also considered as a descent.EXAMPLES:
sage: Permutation([1,4,3,2]).number_of_descents() 2 sage: Permutation([1,4,3,2]).number_of_descents(final_descent=True) 3
>>> from sage.all import * >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).number_of_descents() 2 >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).number_of_descents(final_descent=True) 3
- number_of_fixed_points()[source]¶
Return the number of fixed points of
self
.EXAMPLES:
sage: Permutation([1,3,2,4]).number_of_fixed_points() 2 sage: Permutation([1,2,3,4]).number_of_fixed_points() 4
>>> from sage.all import * >>> Permutation([Integer(1),Integer(3),Integer(2),Integer(4)]).number_of_fixed_points() 2 >>> Permutation([Integer(1),Integer(2),Integer(3),Integer(4)]).number_of_fixed_points() 4
- number_of_idescents(final_descent=False)[source]¶
Return the number of idescents of
self
.See
idescents()
for a definition of idescents.With the
final_descent
option, the last position of a non-empty permutation is also considered as a descent.EXAMPLES:
sage: Permutation([1,4,3,2]).number_of_idescents() 2 sage: Permutation([1,4,3,2]).number_of_idescents(final_descent=True) 3
>>> from sage.all import * >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).number_of_idescents() 2 >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).number_of_idescents(final_descent=True) 3
- number_of_inversions()[source]¶
Return the number of inversions in
self
.An inversion of a permutation is a pair of elements \((i, j)\) with \(i < j\) and \(p(i) > p(j)\).
REFERENCES:
EXAMPLES:
sage: Permutation([3, 2, 4, 1, 5]).number_of_inversions() 4 sage: Permutation([1, 2, 6, 4, 7, 3, 5]).number_of_inversions() 6
>>> from sage.all import * >>> Permutation([Integer(3), Integer(2), Integer(4), Integer(1), Integer(5)]).number_of_inversions() 4 >>> Permutation([Integer(1), Integer(2), Integer(6), Integer(4), Integer(7), Integer(3), Integer(5)]).number_of_inversions() 6
- number_of_noninversions(k)[source]¶
Return the number of
k
-noninversions inself
.If \(k\) is an integer and \(p \in S_n\) is a permutation, then a \(k\)-noninversion in \(p\) is defined as a strictly increasing sequence \((i_1, i_2, \ldots, i_k)\) of elements of \(\{ 1, 2, \ldots, n \}\) satisfying \(p(i_1) < p(i_2) < \cdots < p(i_k)\). (In other words, a \(k\)-noninversion in \(p\) can be regarded as a \(k\)-element subset of \(\{ 1, 2, \ldots, n \}\) on which \(p\) restricts to an increasing map.)
The number of \(k\)-noninversions in \(p\) has been denoted by \(\mathrm{noninv}_k(p)\) in [RSW2011], where conjectures and results regarding this number have been stated.
EXAMPLES:
sage: p = Permutation([3, 2, 4, 1, 5]) sage: p.number_of_noninversions(1) 5 sage: p.number_of_noninversions(2) 6 sage: p.number_of_noninversions(3) 2 sage: p.number_of_noninversions(4) 0 sage: p.number_of_noninversions(5) 0
>>> from sage.all import * >>> p = Permutation([Integer(3), Integer(2), Integer(4), Integer(1), Integer(5)]) >>> p.number_of_noninversions(Integer(1)) 5 >>> p.number_of_noninversions(Integer(2)) 6 >>> p.number_of_noninversions(Integer(3)) 2 >>> p.number_of_noninversions(Integer(4)) 0 >>> p.number_of_noninversions(Integer(5)) 0
The number of \(2\)-noninversions of a permutation \(p \in S_n\) is \(\binom{n}{2}\) minus its number of inversions:
sage: b = binomial(5, 2) # needs sage.symbolic sage: all( x.number_of_noninversions(2) == b - x.number_of_inversions() # needs sage.symbolic ....: for x in Permutations(5) ) True
>>> from sage.all import * >>> b = binomial(Integer(5), Integer(2)) # needs sage.symbolic >>> all( x.number_of_noninversions(Integer(2)) == b - x.number_of_inversions() # needs sage.symbolic ... for x in Permutations(Integer(5)) ) True
We also check some corner cases:
sage: all( x.number_of_noninversions(1) == 5 for x in Permutations(5) ) True sage: all( x.number_of_noninversions(0) == 1 for x in Permutations(5) ) True sage: Permutation([]).number_of_noninversions(1) 0 sage: Permutation([]).number_of_noninversions(0) 1 sage: Permutation([2, 1]).number_of_noninversions(3) 0
>>> from sage.all import * >>> all( x.number_of_noninversions(Integer(1)) == Integer(5) for x in Permutations(Integer(5)) ) True >>> all( x.number_of_noninversions(Integer(0)) == Integer(1) for x in Permutations(Integer(5)) ) True >>> Permutation([]).number_of_noninversions(Integer(1)) 0 >>> Permutation([]).number_of_noninversions(Integer(0)) 1 >>> Permutation([Integer(2), Integer(1)]).number_of_noninversions(Integer(3)) 0
- number_of_nth_roots(n)[source]¶
Return the number of \(n\)-th roots of
self
.An \(n\)-th root of the permutation \(\sigma\) is a permutation \(\gamma\) such that \(\gamma^n = \sigma\).
Note that the number of \(n\)-th roots only depends on the cycle type of
self
.EXAMPLES:
sage: # needs sage.combinat sage: Sigma = Permutations(5).identity() sage: Sigma.number_of_nth_roots(3) 21 sage: Sigma = Permutation('(1, 3)') sage: Sigma.number_of_nth_roots(2) 0
>>> from sage.all import * >>> # needs sage.combinat >>> Sigma = Permutations(Integer(5)).identity() >>> Sigma.number_of_nth_roots(Integer(3)) 21 >>> Sigma = Permutation('(1, 3)') >>> Sigma.number_of_nth_roots(Integer(2)) 0
See also
- number_of_peaks()[source]¶
Return the number of peaks of the permutation
self
.A peak of a permutation \(p\) is an integer \(i\) such that \(p(i-1) < p(i)\) and \(p(i) > p(i+1)\).
EXAMPLES:
sage: Permutation([1,3,2,4,5]).number_of_peaks() 1 sage: Permutation([4,1,3,2,6,5]).number_of_peaks() 2
>>> from sage.all import * >>> Permutation([Integer(1),Integer(3),Integer(2),Integer(4),Integer(5)]).number_of_peaks() 1 >>> Permutation([Integer(4),Integer(1),Integer(3),Integer(2),Integer(6),Integer(5)]).number_of_peaks() 2
- number_of_recoils()[source]¶
Return the number of recoils of the permutation
self
.EXAMPLES:
sage: Permutation([1,4,3,2]).number_of_recoils() 2
>>> from sage.all import * >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).number_of_recoils() 2
- number_of_reduced_words()[source]¶
Return the number of reduced words of
self
without explicitly computing them all.EXAMPLES:
sage: p = Permutation([6,4,2,5,1,8,3,7]) sage: len(p.reduced_words()) == p.number_of_reduced_words() # needs sage.combinat True
>>> from sage.all import * >>> p = Permutation([Integer(6),Integer(4),Integer(2),Integer(5),Integer(1),Integer(8),Integer(3),Integer(7)]) >>> len(p.reduced_words()) == p.number_of_reduced_words() # needs sage.combinat True
- number_of_saliances()[source]¶
Return the number of saliances of
self
.A saliance of a permutation \(p\) is an integer \(i\) such that \(p(i) > p(j)\) for all \(j > i\).
EXAMPLES:
sage: Permutation([2,3,1,5,4]).number_of_saliances() 2 sage: Permutation([5,4,3,2,1]).number_of_saliances() 5
>>> from sage.all import * >>> Permutation([Integer(2),Integer(3),Integer(1),Integer(5),Integer(4)]).number_of_saliances() 2 >>> Permutation([Integer(5),Integer(4),Integer(3),Integer(2),Integer(1)]).number_of_saliances() 5
- order()[source]¶
Return the order of
self
.EXAMPLES:
sage: sigma = Permutation([3,4,1,2,5]) sage: sigma.order() 2 sage: sigma * sigma [1, 2, 3, 4, 5]
>>> from sage.all import * >>> sigma = Permutation([Integer(3),Integer(4),Integer(1),Integer(2),Integer(5)]) >>> sigma.order() 2 >>> sigma * sigma [1, 2, 3, 4, 5]
- pattern_positions(patt)[source]¶
Return the list of positions where the pattern
patt
appears in the permutationself
.EXAMPLES:
sage: Permutation([3,5,1,4,6,2]).pattern_positions([1,3,2]) # needs sage.combinat [[0, 1, 3], [2, 3, 5], [2, 4, 5]]
>>> from sage.all import * >>> Permutation([Integer(3),Integer(5),Integer(1),Integer(4),Integer(6),Integer(2)]).pattern_positions([Integer(1),Integer(3),Integer(2)]) # needs sage.combinat [[0, 1, 3], [2, 3, 5], [2, 4, 5]]
- peaks()[source]¶
Return a list of the peaks of the permutation
self
.A peak of a permutation \(p\) is an integer \(i\) such that \(p(i-1) < p(i)\) and \(p(i) > p(i+1)\).
EXAMPLES:
sage: Permutation([1,3,2,4,5]).peaks() [1] sage: Permutation([4,1,3,2,6,5]).peaks() [2, 4] sage: Permutation([]).peaks() []
>>> from sage.all import * >>> Permutation([Integer(1),Integer(3),Integer(2),Integer(4),Integer(5)]).peaks() [1] >>> Permutation([Integer(4),Integer(1),Integer(3),Integer(2),Integer(6),Integer(5)]).peaks() [2, 4] >>> Permutation([]).peaks() []
- permutation_poset()[source]¶
Return the permutation poset of
self
.The permutation poset of a permutation \(p\) is the poset with vertices \((i, p(i))\) for \(i = 1, 2, \ldots, n\) (where \(n\) is the size of \(p\)) and order inherited from \(\ZZ \times \ZZ\).
EXAMPLES:
sage: # needs sage.combinat sage.graphs sage: Permutation([3,1,5,4,2]).permutation_poset().cover_relations() [[(2, 1), (5, 2)], [(2, 1), (3, 5)], [(2, 1), (4, 4)], [(1, 3), (3, 5)], [(1, 3), (4, 4)]] sage: Permutation([]).permutation_poset().cover_relations() [] sage: Permutation([1,3,2]).permutation_poset().cover_relations() [[(1, 1), (2, 3)], [(1, 1), (3, 2)]] sage: Permutation([1,2]).permutation_poset().cover_relations() [[(1, 1), (2, 2)]] sage: P = Permutation([1,5,2,4,3])
>>> from sage.all import * >>> # needs sage.combinat sage.graphs >>> Permutation([Integer(3),Integer(1),Integer(5),Integer(4),Integer(2)]).permutation_poset().cover_relations() [[(2, 1), (5, 2)], [(2, 1), (3, 5)], [(2, 1), (4, 4)], [(1, 3), (3, 5)], [(1, 3), (4, 4)]] >>> Permutation([]).permutation_poset().cover_relations() [] >>> Permutation([Integer(1),Integer(3),Integer(2)]).permutation_poset().cover_relations() [[(1, 1), (2, 3)], [(1, 1), (3, 2)]] >>> Permutation([Integer(1),Integer(2)]).permutation_poset().cover_relations() [[(1, 1), (2, 2)]] >>> P = Permutation([Integer(1),Integer(5),Integer(2),Integer(4),Integer(3)])
This should hold for any \(P\):
sage: P.permutation_poset().greene_shape() == P.RS_partition() # needs sage.combinat sage.graphs True
>>> from sage.all import * >>> P.permutation_poset().greene_shape() == P.RS_partition() # needs sage.combinat sage.graphs True
- permutohedron_greater(side='right')[source]¶
Return a list of permutations greater than or equal to
self
in the permutohedron order.By default, the computations are done in the right permutohedron. If you pass the option
side='left'
, then they will be done in the left permutohedron.See
permutohedron_lequal()
for the definition of the permutohedron orders.EXAMPLES:
sage: Permutation([4,2,1,3]).permutohedron_greater() [[4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 2, 1]] sage: Permutation([4,2,1,3]).permutohedron_greater(side='left') [[4, 2, 1, 3], [4, 3, 1, 2], [4, 3, 2, 1]]
>>> from sage.all import * >>> Permutation([Integer(4),Integer(2),Integer(1),Integer(3)]).permutohedron_greater() [[4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 2, 1]] >>> Permutation([Integer(4),Integer(2),Integer(1),Integer(3)]).permutohedron_greater(side='left') [[4, 2, 1, 3], [4, 3, 1, 2], [4, 3, 2, 1]]
- permutohedron_join(other, side='right')[source]¶
Return the join of the permutations
self
andother
in the right permutohedron order (or, ifside
is set to'left'
, in the left permutohedron order).The permutohedron orders (see
permutohedron_lequal()
) are lattices; the join operation refers to this lattice structure. In more elementary terms, the join of two permutations \(\pi\) and \(\psi\) in the symmetric group \(S_n\) is the permutation in \(S_n\) whose set of inversion is the transitive closure of the union of the set of inversions of \(\pi\) with the set of inversions of \(\psi\).See also
ALGORITHM:
It is enough to construct the join of any two permutations \(\pi\) and \(\psi\) in \(S_n\) with respect to the right weak order. (The join of \(\pi\) and \(\psi\) with respect to the left weak order is the inverse of the join of \(\pi^{-1}\) and \(\psi^{-1}\) with respect to the right weak order.) Start with an empty list \(l\) (denoted
xs
in the actual code). For \(i = 1, 2, \ldots, n\) (in this order), we insert \(i\) into this list in the rightmost possible position such that any letter in \(\{ 1, 2, ..., i-1 \}\) which appears further right than \(i\) in either \(\pi\) or \(\psi\) (or both) must appear further right than \(i\) in the resulting list. After all numbers are inserted, we are left with a list which is precisely the join of \(\pi\) and \(\psi\) (in one-line notation). This algorithm is due to Markowsky, [Mar1994] (Theorem 1 (a)).AUTHORS:
Viviane Pons and Darij Grinberg, 18 June 2014.
EXAMPLES:
sage: p = Permutation([3,1,2]) sage: q = Permutation([1,3,2]) sage: p.permutohedron_join(q) [3, 1, 2] sage: r = Permutation([2,1,3]) sage: r.permutohedron_join(p) [3, 2, 1]
>>> from sage.all import * >>> p = Permutation([Integer(3),Integer(1),Integer(2)]) >>> q = Permutation([Integer(1),Integer(3),Integer(2)]) >>> p.permutohedron_join(q) [3, 1, 2] >>> r = Permutation([Integer(2),Integer(1),Integer(3)]) >>> r.permutohedron_join(p) [3, 2, 1]
sage: p = Permutation([3,2,4,1]) sage: q = Permutation([4,2,1,3]) sage: p.permutohedron_join(q) [4, 3, 2, 1] sage: r = Permutation([3,1,2,4]) sage: p.permutohedron_join(r) [3, 2, 4, 1] sage: q.permutohedron_join(r) [4, 3, 2, 1] sage: s = Permutation([1,4,2,3]) sage: s.permutohedron_join(r) [4, 3, 1, 2]
>>> from sage.all import * >>> p = Permutation([Integer(3),Integer(2),Integer(4),Integer(1)]) >>> q = Permutation([Integer(4),Integer(2),Integer(1),Integer(3)]) >>> p.permutohedron_join(q) [4, 3, 2, 1] >>> r = Permutation([Integer(3),Integer(1),Integer(2),Integer(4)]) >>> p.permutohedron_join(r) [3, 2, 4, 1] >>> q.permutohedron_join(r) [4, 3, 2, 1] >>> s = Permutation([Integer(1),Integer(4),Integer(2),Integer(3)]) >>> s.permutohedron_join(r) [4, 3, 1, 2]
The universal property of the join operation is satisfied:
sage: def test_uni_join(p, q): ....: j = p.permutohedron_join(q) ....: if not p.permutohedron_lequal(j): ....: return False ....: if not q.permutohedron_lequal(j): ....: return False ....: for r in p.permutohedron_greater(): ....: if q.permutohedron_lequal(r) and not j.permutohedron_lequal(r): ....: return False ....: return True sage: all( test_uni_join(p, q) for p in Permutations(3) for q in Permutations(3) ) True sage: test_uni_join(Permutation([6, 4, 7, 3, 2, 5, 8, 1]), Permutation([7, 3, 1, 2, 5, 4, 6, 8])) True
>>> from sage.all import * >>> def test_uni_join(p, q): ... j = p.permutohedron_join(q) ... if not p.permutohedron_lequal(j): ... return False ... if not q.permutohedron_lequal(j): ... return False ... for r in p.permutohedron_greater(): ... if q.permutohedron_lequal(r) and not j.permutohedron_lequal(r): ... return False ... return True >>> all( test_uni_join(p, q) for p in Permutations(Integer(3)) for q in Permutations(Integer(3)) ) True >>> test_uni_join(Permutation([Integer(6), Integer(4), Integer(7), Integer(3), Integer(2), Integer(5), Integer(8), Integer(1)]), Permutation([Integer(7), Integer(3), Integer(1), Integer(2), Integer(5), Integer(4), Integer(6), Integer(8)])) True
Border cases:
sage: p = Permutation([]) sage: p.permutohedron_join(p) [] sage: p = Permutation([1]) sage: p.permutohedron_join(p) [1]
>>> from sage.all import * >>> p = Permutation([]) >>> p.permutohedron_join(p) [] >>> p = Permutation([Integer(1)]) >>> p.permutohedron_join(p) [1]
The left permutohedron:
sage: p = Permutation([3,1,2]) sage: q = Permutation([1,3,2]) sage: p.permutohedron_join(q, side='left') [3, 2, 1] sage: r = Permutation([2,1,3]) sage: r.permutohedron_join(p, side='left') [3, 1, 2]
>>> from sage.all import * >>> p = Permutation([Integer(3),Integer(1),Integer(2)]) >>> q = Permutation([Integer(1),Integer(3),Integer(2)]) >>> p.permutohedron_join(q, side='left') [3, 2, 1] >>> r = Permutation([Integer(2),Integer(1),Integer(3)]) >>> r.permutohedron_join(p, side='left') [3, 1, 2]
- permutohedron_lequal(p2, side='right')[source]¶
Return
True
ifself
is less or equal top2
in the permutohedron order.By default, the computations are done in the right permutohedron. If you pass the option
side='left'
, then they will be done in the left permutohedron.For every nonnegative integer \(n\), the right (resp. left) permutohedron order (also called the right (resp. left) weak order, or the right (resp. left) weak Bruhat order) is a partial order on the symmetric group \(S_n\). It can be defined in various ways, including the following ones:
Two permutations \(u\) and \(v\) in \(S_n\) satisfy \(u \leq v\) in the right (resp. left) permutohedron order if and only if the (Coxeter) length of the permutation \(v^{-1} \circ u\) (resp. of the permutation \(u \circ v^{-1}\)) equals the length of \(v\) minus the length of \(u\). Here, \(p \circ q\) means the permutation obtained by applying \(q\) first and then \(p\). (Recall that the Coxeter length of a permutation is its number of inversions.)
Two permutations \(u\) and \(v\) in \(S_n\) satisfy \(u \leq v\) in the right (resp. left) permutohedron order if and only if every pair \((i, j)\) of elements of \(\{ 1, 2, \cdots, n \}\) such that \(i < j\) and \(u^{-1}(i) > u^{-1}(j)\) (resp. \(u(i) > u(j)\)) also satisfies \(v^{-1}(i) > v^{-1}(j)\) (resp. \(v(i) > v(j)\)).
A permutation \(v \in S_n\) covers a permutation \(u \in S_n\) in the right (resp. left) permutohedron order if and only if we have \(v = u \circ (i, i + 1)\) (resp. \(v = (i, i + 1) \circ u\)) for some \(i \in \{ 1, 2, \cdots, n - 1 \}\) satisfying \(u(i) < u(i + 1)\) (resp. \(u^{-1}(i) < u^{-1}(i + 1)\)). Here, again, \(p \circ q\) means the permutation obtained by applying \(q\) first and then \(p\).
The right and the left permutohedron order are mutually isomorphic, with the isomorphism being the map sending every permutation to its inverse. Each of these orders endows the symmetric group \(S_n\) with the structure of a graded poset (the rank function being the Coxeter length).
Warning
The permutohedron order is not to be mistaken for the strong Bruhat order (
bruhat_lequal()
), despite both orders being occasionally referred to as the Bruhat order.EXAMPLES:
sage: p = Permutation([3,2,1,4]) sage: p.permutohedron_lequal(Permutation([4,2,1,3])) False sage: p.permutohedron_lequal(Permutation([4,2,1,3]), side='left') True sage: p.permutohedron_lequal(p) True sage: Permutation([2,1,3]).permutohedron_lequal(Permutation([2,3,1])) True sage: Permutation([2,1,3]).permutohedron_lequal(Permutation([3,1,2])) False sage: Permutation([2,1,3]).permutohedron_lequal(Permutation([1,2,3])) False sage: Permutation([1,3,2]).permutohedron_lequal(Permutation([2,1,3])) False sage: Permutation([1,3,2]).permutohedron_lequal(Permutation([2,3,1])) False sage: Permutation([2,3,1]).permutohedron_lequal(Permutation([1,3,2])) False sage: Permutation([2,1,3]).permutohedron_lequal(Permutation([2,3,1]), side='left') False sage: sorted( [len([b for b in Permutations(3) if a.permutohedron_lequal(b)]) ....: for a in Permutations(3)] ) [1, 2, 2, 3, 3, 6] sage: sorted( [len([b for b in Permutations(3) if a.permutohedron_lequal(b, side='left')]) ....: for a in Permutations(3)] ) [1, 2, 2, 3, 3, 6] sage: Permutation([]).permutohedron_lequal(Permutation([])) True
>>> from sage.all import * >>> p = Permutation([Integer(3),Integer(2),Integer(1),Integer(4)]) >>> p.permutohedron_lequal(Permutation([Integer(4),Integer(2),Integer(1),Integer(3)])) False >>> p.permutohedron_lequal(Permutation([Integer(4),Integer(2),Integer(1),Integer(3)]), side='left') True >>> p.permutohedron_lequal(p) True >>> Permutation([Integer(2),Integer(1),Integer(3)]).permutohedron_lequal(Permutation([Integer(2),Integer(3),Integer(1)])) True >>> Permutation([Integer(2),Integer(1),Integer(3)]).permutohedron_lequal(Permutation([Integer(3),Integer(1),Integer(2)])) False >>> Permutation([Integer(2),Integer(1),Integer(3)]).permutohedron_lequal(Permutation([Integer(1),Integer(2),Integer(3)])) False >>> Permutation([Integer(1),Integer(3),Integer(2)]).permutohedron_lequal(Permutation([Integer(2),Integer(1),Integer(3)])) False >>> Permutation([Integer(1),Integer(3),Integer(2)]).permutohedron_lequal(Permutation([Integer(2),Integer(3),Integer(1)])) False >>> Permutation([Integer(2),Integer(3),Integer(1)]).permutohedron_lequal(Permutation([Integer(1),Integer(3),Integer(2)])) False >>> Permutation([Integer(2),Integer(1),Integer(3)]).permutohedron_lequal(Permutation([Integer(2),Integer(3),Integer(1)]), side='left') False >>> sorted( [len([b for b in Permutations(Integer(3)) if a.permutohedron_lequal(b)]) ... for a in Permutations(Integer(3))] ) [1, 2, 2, 3, 3, 6] >>> sorted( [len([b for b in Permutations(Integer(3)) if a.permutohedron_lequal(b, side='left')]) ... for a in Permutations(Integer(3))] ) [1, 2, 2, 3, 3, 6] >>> Permutation([]).permutohedron_lequal(Permutation([])) True
- permutohedron_meet(other, side='right')[source]¶
Return the meet of the permutations
self
andother
in the right permutohedron order (or, ifside
is set to'left'
, in the left permutohedron order).The permutohedron orders (see
permutohedron_lequal()
) are lattices; the meet operation refers to this lattice structure. It is connected to the join operation by the following simple symmetry property: If \(\pi\) and \(\psi\) are two permutations \(\pi\) and \(\psi\) in the symmetric group \(S_n\), and if \(w_0\) denotes the permutation \((n, n-1, \ldots, 1) \in S_n\), then\[\pi \wedge \psi = w_0 \circ ((w_0 \circ \pi) \vee (w_0 \circ \psi)) = ((\pi \circ w_0) \vee (\psi \circ w_0)) \circ w_0\]and
\[\pi \vee \psi = w_0 \circ ((w_0 \circ \pi) \wedge (w_0 \circ \psi)) = ((\pi \circ w_0) \wedge (\psi \circ w_0)) \circ w_0,\]where \(\wedge\) means meet and \(\vee\) means join.
See also
AUTHORS:
Viviane Pons and Darij Grinberg, 18 June 2014.
EXAMPLES:
sage: p = Permutation([3,1,2]) sage: q = Permutation([1,3,2]) sage: p.permutohedron_meet(q) [1, 3, 2] sage: r = Permutation([2,1,3]) sage: r.permutohedron_meet(p) [1, 2, 3]
>>> from sage.all import * >>> p = Permutation([Integer(3),Integer(1),Integer(2)]) >>> q = Permutation([Integer(1),Integer(3),Integer(2)]) >>> p.permutohedron_meet(q) [1, 3, 2] >>> r = Permutation([Integer(2),Integer(1),Integer(3)]) >>> r.permutohedron_meet(p) [1, 2, 3]
sage: p = Permutation([3,2,4,1]) sage: q = Permutation([4,2,1,3]) sage: p.permutohedron_meet(q) [2, 1, 3, 4] sage: r = Permutation([3,1,2,4]) sage: p.permutohedron_meet(r) [3, 1, 2, 4] sage: q.permutohedron_meet(r) [1, 2, 3, 4] sage: s = Permutation([1,4,2,3]) sage: s.permutohedron_meet(r) [1, 2, 3, 4]
>>> from sage.all import * >>> p = Permutation([Integer(3),Integer(2),Integer(4),Integer(1)]) >>> q = Permutation([Integer(4),Integer(2),Integer(1),Integer(3)]) >>> p.permutohedron_meet(q) [2, 1, 3, 4] >>> r = Permutation([Integer(3),Integer(1),Integer(2),Integer(4)]) >>> p.permutohedron_meet(r) [3, 1, 2, 4] >>> q.permutohedron_meet(r) [1, 2, 3, 4] >>> s = Permutation([Integer(1),Integer(4),Integer(2),Integer(3)]) >>> s.permutohedron_meet(r) [1, 2, 3, 4]
The universal property of the meet operation is satisfied:
sage: def test_uni_meet(p, q): ....: m = p.permutohedron_meet(q) ....: if not m.permutohedron_lequal(p): ....: return False ....: if not m.permutohedron_lequal(q): ....: return False ....: for r in p.permutohedron_smaller(): ....: if r.permutohedron_lequal(q) and not r.permutohedron_lequal(m): ....: return False ....: return True sage: all( test_uni_meet(p, q) for p in Permutations(3) for q in Permutations(3) ) True sage: test_uni_meet(Permutation([6, 4, 7, 3, 2, 5, 8, 1]), Permutation([7, 3, 1, 2, 5, 4, 6, 8])) True
>>> from sage.all import * >>> def test_uni_meet(p, q): ... m = p.permutohedron_meet(q) ... if not m.permutohedron_lequal(p): ... return False ... if not m.permutohedron_lequal(q): ... return False ... for r in p.permutohedron_smaller(): ... if r.permutohedron_lequal(q) and not r.permutohedron_lequal(m): ... return False ... return True >>> all( test_uni_meet(p, q) for p in Permutations(Integer(3)) for q in Permutations(Integer(3)) ) True >>> test_uni_meet(Permutation([Integer(6), Integer(4), Integer(7), Integer(3), Integer(2), Integer(5), Integer(8), Integer(1)]), Permutation([Integer(7), Integer(3), Integer(1), Integer(2), Integer(5), Integer(4), Integer(6), Integer(8)])) True
Border cases:
sage: p = Permutation([]) sage: p.permutohedron_meet(p) [] sage: p = Permutation([1]) sage: p.permutohedron_meet(p) [1]
>>> from sage.all import * >>> p = Permutation([]) >>> p.permutohedron_meet(p) [] >>> p = Permutation([Integer(1)]) >>> p.permutohedron_meet(p) [1]
The left permutohedron:
sage: p = Permutation([3,1,2]) sage: q = Permutation([1,3,2]) sage: p.permutohedron_meet(q, side='left') [1, 2, 3] sage: r = Permutation([2,1,3]) sage: r.permutohedron_meet(p, side='left') [2, 1, 3]
>>> from sage.all import * >>> p = Permutation([Integer(3),Integer(1),Integer(2)]) >>> q = Permutation([Integer(1),Integer(3),Integer(2)]) >>> p.permutohedron_meet(q, side='left') [1, 2, 3] >>> r = Permutation([Integer(2),Integer(1),Integer(3)]) >>> r.permutohedron_meet(p, side='left') [2, 1, 3]
- permutohedron_pred(side='right')[source]¶
Return a list of the permutations strictly smaller than
self
in the permutohedron order such that there is no permutation between any of those andself
.By default, the computations are done in the right permutohedron. If you pass the option
side='left'
, then they will be done in the left permutohedron.See
permutohedron_lequal()
for the definition of the permutohedron orders.EXAMPLES:
sage: p = Permutation([4,2,1,3]) sage: p.permutohedron_pred() [[2, 4, 1, 3], [4, 1, 2, 3]] sage: p.permutohedron_pred(side='left') [[4, 1, 2, 3], [3, 2, 1, 4]]
>>> from sage.all import * >>> p = Permutation([Integer(4),Integer(2),Integer(1),Integer(3)]) >>> p.permutohedron_pred() [[2, 4, 1, 3], [4, 1, 2, 3]] >>> p.permutohedron_pred(side='left') [[4, 1, 2, 3], [3, 2, 1, 4]]
- permutohedron_smaller(side='right')[source]¶
Return a list of permutations smaller than or equal to
self
in the permutohedron order.By default, the computations are done in the right permutohedron. If you pass the option
side='left'
, then they will be done in the left permutohedron.See
permutohedron_lequal()
for the definition of the permutohedron orders.EXAMPLES:
sage: Permutation([4,2,1,3]).permutohedron_smaller() [[1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3], [2, 1, 3, 4], [2, 1, 4, 3], [2, 4, 1, 3], [4, 1, 2, 3], [4, 2, 1, 3]]
>>> from sage.all import * >>> Permutation([Integer(4),Integer(2),Integer(1),Integer(3)]).permutohedron_smaller() [[1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3], [2, 1, 3, 4], [2, 1, 4, 3], [2, 4, 1, 3], [4, 1, 2, 3], [4, 2, 1, 3]]
sage: Permutation([4,2,1,3]).permutohedron_smaller(side='left') [[1, 2, 3, 4], [1, 3, 2, 4], [2, 1, 3, 4], [2, 3, 1, 4], [3, 1, 2, 4], [3, 2, 1, 4], [4, 1, 2, 3], [4, 2, 1, 3]]
>>> from sage.all import * >>> Permutation([Integer(4),Integer(2),Integer(1),Integer(3)]).permutohedron_smaller(side='left') [[1, 2, 3, 4], [1, 3, 2, 4], [2, 1, 3, 4], [2, 3, 1, 4], [3, 1, 2, 4], [3, 2, 1, 4], [4, 1, 2, 3], [4, 2, 1, 3]]
- permutohedron_succ(side='right')[source]¶
Return a list of the permutations strictly greater than
self
in the permutohedron order such that there is no permutation between any of those andself
.By default, the computations are done in the right permutohedron. If you pass the option
side='left'
, then they will be done in the left permutohedron.See
permutohedron_lequal()
for the definition of the permutohedron orders.EXAMPLES:
sage: p = Permutation([4,2,1,3]) sage: p.permutohedron_succ() [[4, 2, 3, 1]] sage: p.permutohedron_succ(side='left') [[4, 3, 1, 2]]
>>> from sage.all import * >>> p = Permutation([Integer(4),Integer(2),Integer(1),Integer(3)]) >>> p.permutohedron_succ() [[4, 2, 3, 1]] >>> p.permutohedron_succ(side='left') [[4, 3, 1, 2]]
- prev()[source]¶
Return the permutation that comes directly before
self
in lexicographic order on the symmetric group containingself
. Ifself
is the first permutation, then it returnsFalse
. Does not support the Steinhaus-Johnson-Trotter algorithm for the moment.EXAMPLES:
sage: p = Permutation([1,2,3]) sage: p.prev() False sage: p = Permutation([1,3,2]) sage: p.prev() [1, 2, 3]
>>> from sage.all import * >>> p = Permutation([Integer(1),Integer(2),Integer(3)]) >>> p.prev() False >>> p = Permutation([Integer(1),Integer(3),Integer(2)]) >>> p.prev() [1, 2, 3]
Todo
Implement the previous permutation for the Steinhaus-Johnson-Trotter algorithm.
- rank()[source]¶
Return the rank of
self
in the lexicographic ordering on the symmetric group to whichself
belongs.EXAMPLES:
sage: Permutation([1,2,3]).rank() 0 sage: Permutation([1, 2, 4, 6, 3, 5]).rank() 10 sage: perms = Permutations(6).list() sage: [p.rank() for p in perms] == list(range(factorial(6))) True
>>> from sage.all import * >>> Permutation([Integer(1),Integer(2),Integer(3)]).rank() 0 >>> Permutation([Integer(1), Integer(2), Integer(4), Integer(6), Integer(3), Integer(5)]).rank() 10 >>> perms = Permutations(Integer(6)).list() >>> [p.rank() for p in perms] == list(range(factorial(Integer(6)))) True
- recoils()[source]¶
Return the list of the positions of the recoils of
self
.A recoil of a permutation \(p\) is an integer \(i\) such that \(i+1\) appears to the left of \(i\) in \(p\). Here, the positions are being counted starting at \(0\). (Note that it is the positions, not the recoils themselves, which are being listed.)
EXAMPLES:
sage: Permutation([1,4,3,2]).recoils() [2, 3] sage: Permutation([]).recoils() []
>>> from sage.all import * >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).recoils() [2, 3] >>> Permutation([]).recoils() []
- recoils_composition()[source]¶
Return the recoils composition of
self
.The recoils composition of a permutation \(p \in S_n\) is the composition of \(n\) whose descent set is the set of the recoils of \(p\) (not their positions). In other words, this is the descents composition of \(p^{-1}\).
EXAMPLES:
sage: Permutation([1,3,2,4]).recoils_composition() [2, 2] sage: Permutation([]).recoils_composition() []
>>> from sage.all import * >>> Permutation([Integer(1),Integer(3),Integer(2),Integer(4)]).recoils_composition() [2, 2] >>> Permutation([]).recoils_composition() []
- reduced_word()[source]¶
Return a reduced word of the permutation
self
.See
reduced_words()
for the definition of reduced words and a way to compute them all.Warning
This does not respect the multiplication convention.
EXAMPLES:
sage: Permutation([3,5,4,6,2,1]).reduced_word() [2, 1, 4, 3, 2, 4, 3, 5, 4, 5] Permutation([1]).reduced_word_lexmin() [] Permutation([]).reduced_word_lexmin() []
>>> from sage.all import * >>> Permutation([Integer(3),Integer(5),Integer(4),Integer(6),Integer(2),Integer(1)]).reduced_word() [2, 1, 4, 3, 2, 4, 3, 5, 4, 5] Permutation([1]).reduced_word_lexmin() [] Permutation([]).reduced_word_lexmin() []
- reduced_word_lexmin()[source]¶
Return a lexicographically minimal reduced word of the permutation
self
.See
reduced_words()
for the definition of reduced words and a way to compute them all.EXAMPLES:
sage: Permutation([3,4,2,1]).reduced_word_lexmin() [1, 2, 1, 3, 2] Permutation([1]).reduced_word_lexmin() [] Permutation([]).reduced_word_lexmin() []
>>> from sage.all import * >>> Permutation([Integer(3),Integer(4),Integer(2),Integer(1)]).reduced_word_lexmin() [1, 2, 1, 3, 2] Permutation([1]).reduced_word_lexmin() [] Permutation([]).reduced_word_lexmin() []
- reduced_words()[source]¶
Return a list of the reduced words of
self
.The notion of a reduced word is based on the well-known fact that every permutation can be written as a product of adjacent transpositions. In more detail: If \(n\) is a nonnegative integer, we can define the transpositions \(s_i = (i, i+1) \in S_n\) for all \(i \in \{ 1, 2, \ldots, n-1 \}\), and every \(p \in S_n\) can then be written as a product \(s_{i_1} s_{i_2} \cdots s_{i_k}\) for some sequence \((i_1, i_2, \ldots, i_k)\) of elements of \(\{ 1, 2, \ldots, n-1 \}\) (here \(\{ 1, 2, \ldots, n-1 \}\) denotes the empty set when \(n \leq 1\)). Fixing a \(p\), the sequences \((i_1, i_2, \ldots, i_k)\) of smallest length satisfying \(p = s_{i_1} s_{i_2} \cdots s_{i_k}\) are called the reduced words of \(p\). (Their length is the Coxeter length of \(p\), and can be computed using
length()
.)Note that the product of permutations is defined here in such a way that \((pq)(i) = p(q(i))\) for all permutations \(p\) and \(q\) and each \(i \in \{ 1, 2, \ldots, n \}\) (this is the same convention as in
left_action_product()
, but not the default semantics of the \(*\) operator on permutations in Sage). Thus, for instance, \(s_2 s_1\) is the permutation obtained by first transposing \(1\) with \(2\) and then transposing \(2\) with \(3\).See also
EXAMPLES:
sage: Permutation([2,1,3]).reduced_words() [[1]] sage: Permutation([3,1,2]).reduced_words() [[2, 1]] sage: Permutation([3,2,1]).reduced_words() [[1, 2, 1], [2, 1, 2]] sage: Permutation([3,2,4,1]).reduced_words() [[1, 2, 3, 1], [1, 2, 1, 3], [2, 1, 2, 3]] Permutation([1]).reduced_words() [[]] Permutation([]).reduced_words() [[]]
>>> from sage.all import * >>> Permutation([Integer(2),Integer(1),Integer(3)]).reduced_words() [[1]] >>> Permutation([Integer(3),Integer(1),Integer(2)]).reduced_words() [[2, 1]] >>> Permutation([Integer(3),Integer(2),Integer(1)]).reduced_words() [[1, 2, 1], [2, 1, 2]] >>> Permutation([Integer(3),Integer(2),Integer(4),Integer(1)]).reduced_words() [[1, 2, 3, 1], [1, 2, 1, 3], [2, 1, 2, 3]] Permutation([1]).reduced_words() [[]] Permutation([]).reduced_words() [[]]
- reduced_words_iterator()[source]¶
Return an iterator for the reduced words of
self
.EXAMPLES:
sage: next(Permutation([5,2,3,4,1]).reduced_words_iterator()) [1, 2, 3, 4, 3, 2, 1]
>>> from sage.all import * >>> next(Permutation([Integer(5),Integer(2),Integer(3),Integer(4),Integer(1)]).reduced_words_iterator()) [1, 2, 3, 4, 3, 2, 1]
- remove_extra_fixed_points()[source]¶
Return the permutation obtained by removing any fixed points at the end of
self
.However, return
[1]
rather than[]
ifself
is the identity permutation.This is mostly a helper method for
sage.combinat.schubert_polynomial
, where it is used to normalize finitary permutations of \(\{1,2,3,\ldots\}\).EXAMPLES:
sage: Permutation([2,1,3]).remove_extra_fixed_points() [2, 1] sage: Permutation([1,2,3,4]).remove_extra_fixed_points() [1] sage: Permutation([2,1]).remove_extra_fixed_points() [2, 1] sage: Permutation([]).remove_extra_fixed_points() [1]
>>> from sage.all import * >>> Permutation([Integer(2),Integer(1),Integer(3)]).remove_extra_fixed_points() [2, 1] >>> Permutation([Integer(1),Integer(2),Integer(3),Integer(4)]).remove_extra_fixed_points() [1] >>> Permutation([Integer(2),Integer(1)]).remove_extra_fixed_points() [2, 1] >>> Permutation([]).remove_extra_fixed_points() [1]
See also
- retract_direct_product(m)[source]¶
Return the direct-product retract of the permutation
self
\(\in S_n\) to \(S_m\), where \(m \leq n\). If this retract is undefined, thenNone
is returned.If \(p \in S_n\) is a permutation, and \(m\) is a nonnegative integer less or equal to \(n\), then the direct-product retract of \(p\) to \(S_m\) is defined only if \(p([m]) = [m]\), where \([m]\) denotes the interval \(\{1, 2, \ldots, m\}\). In this case, it is defined as the permutation written \((p(1), p(2), \ldots, p(m))\) in one-line notation.
EXAMPLES:
sage: Permutation([4,1,2,3,5]).retract_direct_product(4) [4, 1, 2, 3] sage: Permutation([4,1,2,3,5]).retract_direct_product(3) sage: Permutation([1,4,2,3,6,5]).retract_direct_product(5) sage: Permutation([1,4,2,3,6,5]).retract_direct_product(4) [1, 4, 2, 3] sage: Permutation([1,4,2,3,6,5]).retract_direct_product(3) sage: Permutation([1,4,2,3,6,5]).retract_direct_product(2) sage: Permutation([1,4,2,3,6,5]).retract_direct_product(1) [1] sage: Permutation([1,4,2,3,6,5]).retract_direct_product(0) [] sage: all( p.retract_direct_product(3) == p for p in Permutations(3) ) True
>>> from sage.all import * >>> Permutation([Integer(4),Integer(1),Integer(2),Integer(3),Integer(5)]).retract_direct_product(Integer(4)) [4, 1, 2, 3] >>> Permutation([Integer(4),Integer(1),Integer(2),Integer(3),Integer(5)]).retract_direct_product(Integer(3)) >>> Permutation([Integer(1),Integer(4),Integer(2),Integer(3),Integer(6),Integer(5)]).retract_direct_product(Integer(5)) >>> Permutation([Integer(1),Integer(4),Integer(2),Integer(3),Integer(6),Integer(5)]).retract_direct_product(Integer(4)) [1, 4, 2, 3] >>> Permutation([Integer(1),Integer(4),Integer(2),Integer(3),Integer(6),Integer(5)]).retract_direct_product(Integer(3)) >>> Permutation([Integer(1),Integer(4),Integer(2),Integer(3),Integer(6),Integer(5)]).retract_direct_product(Integer(2)) >>> Permutation([Integer(1),Integer(4),Integer(2),Integer(3),Integer(6),Integer(5)]).retract_direct_product(Integer(1)) [1] >>> Permutation([Integer(1),Integer(4),Integer(2),Integer(3),Integer(6),Integer(5)]).retract_direct_product(Integer(0)) [] >>> all( p.retract_direct_product(Integer(3)) == p for p in Permutations(Integer(3)) ) True
See also
- retract_okounkov_vershik(m)[source]¶
Return the Okounkov-Vershik retract of the permutation
self
\(\in S_n\) to \(S_m\), where \(m \leq n\).If \(p \in S_n\) is a permutation, and \(m\) is a nonnegative integer less or equal to \(n\), then the Okounkov-Vershik retract of \(p\) to \(S_m\) is defined as the permutation in \(S_m\) which sends every \(i \in \{1, 2, \ldots, m\}\) to \(p^{k_i}(i)\), where \(k_i\) is the smallest positive integer \(k\) satisfying \(p^k(i) \leq m\).
In other words, the Okounkov-Vershik retract of \(p\) is the permutation whose disjoint cycle decomposition is obtained by removing all letters strictly greater than \(m\) from the decomposition of \(p\) into disjoint cycles (and removing all cycles which are emptied in the process).
When \(m = n-1\), the Okounkov-Vershik retract (as a map \(S_n \to S_{n-1}\)) is the map \(\widetilde{p}_n\) introduced in Section 7 of [VO2005], and appears as (3.20) in [CST2010]. In the general case, the Okounkov-Vershik retract of a permutation in \(S_n\) to \(S_m\) can be obtained by first taking its Okounkov-Vershik retract to \(S_{n-1}\), then that of the resulting permutation to \(S_{n-2}\), etc. until arriving in \(S_m\).
EXAMPLES:
sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(4) [4, 1, 2, 3] sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(3) [3, 1, 2] sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(2) [2, 1] sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(1) [1] sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(0) [] sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(5) [1, 4, 2, 3, 5] sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(4) [1, 4, 2, 3] sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(3) [1, 3, 2] sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(2) [1, 2] sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(1) [1] sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(0) [] sage: Permutation([6,5,4,3,2,1]).retract_okounkov_vershik(5) [1, 5, 4, 3, 2] sage: Permutation([6,5,4,3,2,1]).retract_okounkov_vershik(4) [1, 2, 4, 3] sage: Permutation([1,5,2,6,3,7,4,8]).retract_okounkov_vershik(4) [1, 3, 2, 4] sage: all( p.retract_direct_product(3) == p for p in Permutations(3) ) True
>>> from sage.all import * >>> Permutation([Integer(4),Integer(1),Integer(2),Integer(3),Integer(5)]).retract_okounkov_vershik(Integer(4)) [4, 1, 2, 3] >>> Permutation([Integer(4),Integer(1),Integer(2),Integer(3),Integer(5)]).retract_okounkov_vershik(Integer(3)) [3, 1, 2] >>> Permutation([Integer(4),Integer(1),Integer(2),Integer(3),Integer(5)]).retract_okounkov_vershik(Integer(2)) [2, 1] >>> Permutation([Integer(4),Integer(1),Integer(2),Integer(3),Integer(5)]).retract_okounkov_vershik(Integer(1)) [1] >>> Permutation([Integer(4),Integer(1),Integer(2),Integer(3),Integer(5)]).retract_okounkov_vershik(Integer(0)) [] >>> Permutation([Integer(1),Integer(4),Integer(2),Integer(3),Integer(6),Integer(5)]).retract_okounkov_vershik(Integer(5)) [1, 4, 2, 3, 5] >>> Permutation([Integer(1),Integer(4),Integer(2),Integer(3),Integer(6),Integer(5)]).retract_okounkov_vershik(Integer(4)) [1, 4, 2, 3] >>> Permutation([Integer(1),Integer(4),Integer(2),Integer(3),Integer(6),Integer(5)]).retract_okounkov_vershik(Integer(3)) [1, 3, 2] >>> Permutation([Integer(1),Integer(4),Integer(2),Integer(3),Integer(6),Integer(5)]).retract_okounkov_vershik(Integer(2)) [1, 2] >>> Permutation([Integer(1),Integer(4),Integer(2),Integer(3),Integer(6),Integer(5)]).retract_okounkov_vershik(Integer(1)) [1] >>> Permutation([Integer(1),Integer(4),Integer(2),Integer(3),Integer(6),Integer(5)]).retract_okounkov_vershik(Integer(0)) [] >>> Permutation([Integer(6),Integer(5),Integer(4),Integer(3),Integer(2),Integer(1)]).retract_okounkov_vershik(Integer(5)) [1, 5, 4, 3, 2] >>> Permutation([Integer(6),Integer(5),Integer(4),Integer(3),Integer(2),Integer(1)]).retract_okounkov_vershik(Integer(4)) [1, 2, 4, 3] >>> Permutation([Integer(1),Integer(5),Integer(2),Integer(6),Integer(3),Integer(7),Integer(4),Integer(8)]).retract_okounkov_vershik(Integer(4)) [1, 3, 2, 4] >>> all( p.retract_direct_product(Integer(3)) == p for p in Permutations(Integer(3)) ) True
See also
- retract_plain(m)[source]¶
Return the plain retract of the permutation
self
in \(S_n\) to \(S_m\), where \(m \leq n\). If this retract is undefined, thenNone
is returned.If \(p \in S_n\) is a permutation, and \(m\) is a nonnegative integer less or equal to \(n\), then the plain retract of \(p\) to \(S_m\) is defined only if every \(i > m\) satisfies \(p(i) = i\). In this case, it is defined as the permutation written \((p(1), p(2), \ldots, p(m))\) in one-line notation.
EXAMPLES:
sage: Permutation([4,1,2,3,5]).retract_plain(4) [4, 1, 2, 3] sage: Permutation([4,1,2,3,5]).retract_plain(3) sage: Permutation([1,3,2,4,5,6]).retract_plain(3) [1, 3, 2] sage: Permutation([1,3,2,4,5,6]).retract_plain(2) sage: Permutation([1,2,3,4,5]).retract_plain(1) [1] sage: Permutation([1,2,3,4,5]).retract_plain(0) [] sage: all( p.retract_plain(3) == p for p in Permutations(3) ) True
>>> from sage.all import * >>> Permutation([Integer(4),Integer(1),Integer(2),Integer(3),Integer(5)]).retract_plain(Integer(4)) [4, 1, 2, 3] >>> Permutation([Integer(4),Integer(1),Integer(2),Integer(3),Integer(5)]).retract_plain(Integer(3)) >>> Permutation([Integer(1),Integer(3),Integer(2),Integer(4),Integer(5),Integer(6)]).retract_plain(Integer(3)) [1, 3, 2] >>> Permutation([Integer(1),Integer(3),Integer(2),Integer(4),Integer(5),Integer(6)]).retract_plain(Integer(2)) >>> Permutation([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]).retract_plain(Integer(1)) [1] >>> Permutation([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]).retract_plain(Integer(0)) [] >>> all( p.retract_plain(Integer(3)) == p for p in Permutations(Integer(3)) ) True
- reverse()[source]¶
Return the permutation obtained by reversing the list.
EXAMPLES:
sage: Permutation([3,4,1,2]).reverse() [2, 1, 4, 3] sage: Permutation([1,2,3,4,5]).reverse() [5, 4, 3, 2, 1]
>>> from sage.all import * >>> Permutation([Integer(3),Integer(4),Integer(1),Integer(2)]).reverse() [2, 1, 4, 3] >>> Permutation([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]).reverse() [5, 4, 3, 2, 1]
- right_action_product(rp)[source]¶
Return the permutation obtained by composing
self
withrp
in such an order thatself
is applied first andrp
is applied afterwards.This is usually denoted by either
self * rp
orrp * self
depending on the conventions used by the author. If the value of a permutation \(p \in S_n\) on an integer \(i \in \{ 1, 2, \cdots, n \}\) is denoted by \(p(i)\), then this should be denoted byrp * self
in order to have associativity (i.e., in order to have \((p \cdot q)(i) = p(q(i))\) for all \(p\), \(q\) and \(i\)). If, on the other hand, the value of a permutation \(p \in S_n\) on an integer \(i \in \{ 1, 2, \cdots, n \}\) is denoted by \(i^p\), then this should be denoted byself * rp
in order to have associativity (i.e., in order to have \(i^{p \cdot q} = (i^p)^q\) for all \(p\), \(q\) and \(i\)).EXAMPLES:
sage: p = Permutation([2,1,3]) sage: q = Permutation([3,1,2]) sage: p.right_action_product(q) [1, 3, 2] sage: q.right_action_product(p) [3, 2, 1]
>>> from sage.all import * >>> p = Permutation([Integer(2),Integer(1),Integer(3)]) >>> q = Permutation([Integer(3),Integer(1),Integer(2)]) >>> p.right_action_product(q) [1, 3, 2] >>> q.right_action_product(p) [3, 2, 1]
- right_permutohedron_interval(other)[source]¶
Return the list of the permutations belonging to the right permutohedron interval where
self
is the minimal element andother
the maximal element.See
permutohedron_lequal()
for the definition of the permutohedron orders.EXAMPLES:
sage: p = Permutation([2, 1, 4, 5, 3]); q = Permutation([2, 5, 4, 1, 3]) sage: p.right_permutohedron_interval(q) # needs sage.graphs sage.modules [[2, 4, 5, 1, 3], [2, 4, 1, 5, 3], [2, 1, 4, 5, 3], [2, 1, 5, 4, 3], [2, 5, 1, 4, 3], [2, 5, 4, 1, 3]]
>>> from sage.all import * >>> p = Permutation([Integer(2), Integer(1), Integer(4), Integer(5), Integer(3)]); q = Permutation([Integer(2), Integer(5), Integer(4), Integer(1), Integer(3)]) >>> p.right_permutohedron_interval(q) # needs sage.graphs sage.modules [[2, 4, 5, 1, 3], [2, 4, 1, 5, 3], [2, 1, 4, 5, 3], [2, 1, 5, 4, 3], [2, 5, 1, 4, 3], [2, 5, 4, 1, 3]]
- right_permutohedron_interval_iterator(other)[source]¶
Return an iterator on the permutations (represented as integer lists) belonging to the right permutohedron interval where
self
is the minimal element andother
the maximal element.See
permutohedron_lequal()
for the definition of the permutohedron orders.EXAMPLES:
sage: p = Permutation([2, 1, 4, 5, 3]); q = Permutation([2, 5, 4, 1, 3]) sage: p.right_permutohedron_interval(q) # indirect doctest # needs sage.graphs sage.modules [[2, 4, 5, 1, 3], [2, 4, 1, 5, 3], [2, 1, 4, 5, 3], [2, 1, 5, 4, 3], [2, 5, 1, 4, 3], [2, 5, 4, 1, 3]]
>>> from sage.all import * >>> p = Permutation([Integer(2), Integer(1), Integer(4), Integer(5), Integer(3)]); q = Permutation([Integer(2), Integer(5), Integer(4), Integer(1), Integer(3)]) >>> p.right_permutohedron_interval(q) # indirect doctest # needs sage.graphs sage.modules [[2, 4, 5, 1, 3], [2, 4, 1, 5, 3], [2, 1, 4, 5, 3], [2, 1, 5, 4, 3], [2, 5, 1, 4, 3], [2, 5, 4, 1, 3]]
- right_tableau()[source]¶
Return the right standard tableau after performing the RSK algorithm on
self
.EXAMPLES:
sage: Permutation([1,4,3,2]).right_tableau() # needs sage.combinat [[1, 2], [3], [4]]
>>> from sage.all import * >>> Permutation([Integer(1),Integer(4),Integer(3),Integer(2)]).right_tableau() # needs sage.combinat [[1, 2], [3], [4]]
- robinson_schensted()[source]¶
Return the pair of standard tableaux obtained by running the Robinson-Schensted algorithm on
self
.This can also be done by running
RSK()
onself
(with the optional argumentcheck_standard=True
to return standard Young tableaux).EXAMPLES:
sage: Permutation([6,2,3,1,7,5,4]).robinson_schensted() # needs sage.combinat [[[1, 3, 4], [2, 5], [6, 7]], [[1, 3, 5], [2, 6], [4, 7]]]
>>> from sage.all import * >>> Permutation([Integer(6),Integer(2),Integer(3),Integer(1),Integer(7),Integer(5),Integer(4)]).robinson_schensted() # needs sage.combinat [[[1, 3, 4], [2, 5], [6, 7]], [[1, 3, 5], [2, 6], [4, 7]]]
- rothe_diagram()[source]¶
Return the Rothe diagram of
self
.EXAMPLES:
sage: p = Permutation([4,2,1,3]) sage: D = p.rothe_diagram(); D # needs sage.combinat [(0, 0), (0, 1), (0, 2), (1, 0)] sage: D.pp() # needs sage.combinat O O O . O . . . . . . . . . . .
>>> from sage.all import * >>> p = Permutation([Integer(4),Integer(2),Integer(1),Integer(3)]) >>> D = p.rothe_diagram(); D # needs sage.combinat [(0, 0), (0, 1), (0, 2), (1, 0)] >>> D.pp() # needs sage.combinat O O O . O . . . . . . . . . . .
- runs(as_tuple=False)[source]¶
Return a list of the runs in the nonempty permutation
self
.A run in a permutation is defined to be a maximal (with respect to inclusion) nonempty increasing substring (i. e., contiguous subsequence). For instance, the runs in the permutation
[6,1,7,3,4,5,2]
are[6]
,[1,7]
,[3,4,5]
and[2]
.Runs in an empty permutation are not defined.
INPUT:
as_tuple
– boolean (default:False
); choice of output format
OUTPUT: list of lists or a tuple of tuples
REFERENCES:
EXAMPLES:
sage: Permutation([1,2,3,4]).runs() [[1, 2, 3, 4]] sage: Permutation([4,3,2,1]).runs() [[4], [3], [2], [1]] sage: Permutation([2,4,1,3]).runs() [[2, 4], [1, 3]] sage: Permutation([1]).runs() [[1]]
>>> from sage.all import * >>> Permutation([Integer(1),Integer(2),Integer(3),Integer(4)]).runs() [[1, 2, 3, 4]] >>> Permutation([Integer(4),Integer(3),Integer(2),Integer(1)]).runs() [[4], [3], [2], [1]] >>> Permutation([Integer(2),Integer(4),Integer(1),Integer(3)]).runs() [[2, 4], [1, 3]] >>> Permutation([Integer(1)]).runs() [[1]]
The example from above:
sage: Permutation([6,1,7,3,4,5,2]).runs() [[6], [1, 7], [3, 4, 5], [2]] sage: Permutation([6,1,7,3,4,5,2]).runs(as_tuple=True) ((6,), (1, 7), (3, 4, 5), (2,))
>>> from sage.all import * >>> Permutation([Integer(6),Integer(1),Integer(7),Integer(3),Integer(4),Integer(5),Integer(2)]).runs() [[6], [1, 7], [3, 4, 5], [2]] >>> Permutation([Integer(6),Integer(1),Integer(7),Integer(3),Integer(4),Integer(5),Integer(2)]).runs(as_tuple=True) ((6,), (1, 7), (3, 4, 5), (2,))
The number of runs in a nonempty permutation equals its number of descents plus 1:
sage: all( len(p.runs()) == p.number_of_descents() + 1 ....: for p in Permutations(6) ) True
>>> from sage.all import * >>> all( len(p.runs()) == p.number_of_descents() + Integer(1) ... for p in Permutations(Integer(6)) ) True
- saliances()[source]¶
Return a list of the saliances of the permutation
self
.A saliance of a permutation \(p\) is an integer \(i\) such that \(p(i) > p(j)\) for all \(j > i\).
EXAMPLES:
sage: Permutation([2,3,1,5,4]).saliances() [3, 4] sage: Permutation([5,4,3,2,1]).saliances() [0, 1, 2, 3, 4]
>>> from sage.all import * >>> Permutation([Integer(2),Integer(3),Integer(1),Integer(5),Integer(4)]).saliances() [3, 4] >>> Permutation([Integer(5),Integer(4),Integer(3),Integer(2),Integer(1)]).saliances() [0, 1, 2, 3, 4]
- shifted_concatenation(other, side='right')[source]¶
Return the right (or left) shifted concatenation of
self
with a permutationother
. These operations are also known as the Loday-Ronco over and under operations.INPUT:
other
– a permutation, a list, a tuple, or any iterable representing a permutationside
– string (default:'right'
);'left'
or'right'
OUTPUT:
If
side
is'right'
, the method returns the permutation obtained by concatenatingself
with the letters ofother
incremented by the size ofself
. This is what is calledside / other
in [LR0102066], and denoted as the “over” operation. Otherwise, i. e., whenside
is'left'
, the method returns the permutation obtained by concatenating the letters ofother
incremented by the size ofself
withself
. This is what is calledside \ other
in [LR0102066] (which seems to use the \((\sigma \pi)(i) = \pi(\sigma(i))\) convention for the product of permutations).EXAMPLES:
sage: Permutation([]).shifted_concatenation(Permutation([]), "right") [] sage: Permutation([]).shifted_concatenation(Permutation([]), "left") [] sage: Permutation([2, 4, 1, 3]).shifted_concatenation(Permutation([3, 1, 2]), "right") [2, 4, 1, 3, 7, 5, 6] sage: Permutation([2, 4, 1, 3]).shifted_concatenation(Permutation([3, 1, 2]), "left") [7, 5, 6, 2, 4, 1, 3]
>>> from sage.all import * >>> Permutation([]).shifted_concatenation(Permutation([]), "right") [] >>> Permutation([]).shifted_concatenation(Permutation([]), "left") [] >>> Permutation([Integer(2), Integer(4), Integer(1), Integer(3)]).shifted_concatenation(Permutation([Integer(3), Integer(1), Integer(2)]), "right") [2, 4, 1, 3, 7, 5, 6] >>> Permutation([Integer(2), Integer(4), Integer(1), Integer(3)]).shifted_concatenation(Permutation([Integer(3), Integer(1), Integer(2)]), "left") [7, 5, 6, 2, 4, 1, 3]
- shifted_shuffle(other)[source]¶
Return the shifted shuffle of two permutations
self
andother
.INPUT:
other
– a permutation, a list, a tuple, or any iterable representing a permutation
OUTPUT:
The list of the permutations appearing in the shifted shuffle of the permutations
self
andother
.EXAMPLES:
sage: # needs sage.graphs sage.modules sage: Permutation([]).shifted_shuffle(Permutation([])) [[]] sage: Permutation([1, 2, 3]).shifted_shuffle(Permutation([1])) [[4, 1, 2, 3], [1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3]] sage: Permutation([1, 2]).shifted_shuffle(Permutation([2, 1])) [[4, 1, 3, 2], [4, 3, 1, 2], [1, 4, 3, 2], [1, 4, 2, 3], [1, 2, 4, 3], [4, 1, 2, 3]] sage: Permutation([1]).shifted_shuffle([1]) [[2, 1], [1, 2]] sage: p = Permutation([3, 1, 5, 4, 2]) sage: len(p.shifted_shuffle(Permutation([2, 1, 4, 3]))) 126
>>> from sage.all import * >>> # needs sage.graphs sage.modules >>> Permutation([]).shifted_shuffle(Permutation([])) [[]] >>> Permutation([Integer(1), Integer(2), Integer(3)]).shifted_shuffle(Permutation([Integer(1)])) [[4, 1, 2, 3], [1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3]] >>> Permutation([Integer(1), Integer(2)]).shifted_shuffle(Permutation([Integer(2), Integer(1)])) [[4, 1, 3, 2], [4, 3, 1, 2], [1, 4, 3, 2], [1, 4, 2, 3], [1, 2, 4, 3], [4, 1, 2, 3]] >>> Permutation([Integer(1)]).shifted_shuffle([Integer(1)]) [[2, 1], [1, 2]] >>> p = Permutation([Integer(3), Integer(1), Integer(5), Integer(4), Integer(2)]) >>> len(p.shifted_shuffle(Permutation([Integer(2), Integer(1), Integer(4), Integer(3)]))) 126
The shifted shuffle product is associative. We can test this on an admittedly toy example:
sage: all( all( all( sorted(flatten([abs.shifted_shuffle(c) # needs sage.graphs sage.modules ....: for abs in a.shifted_shuffle(b)])) ....: == sorted(flatten([a.shifted_shuffle(bcs) ....: for bcs in b.shifted_shuffle(c)])) ....: for c in Permutations(2) ) ....: for b in Permutations(2) ) ....: for a in Permutations(2) ) True
>>> from sage.all import * >>> all( all( all( sorted(flatten([abs.shifted_shuffle(c) # needs sage.graphs sage.modules ... for abs in a.shifted_shuffle(b)])) ... == sorted(flatten([a.shifted_shuffle(bcs) ... for bcs in b.shifted_shuffle(c)])) ... for c in Permutations(Integer(2)) ) ... for b in Permutations(Integer(2)) ) ... for a in Permutations(Integer(2)) ) True
The
shifted_shuffle
method on permutations gives the same permutations as theshifted_shuffle
method on words (but is faster):sage: all( all( sorted(p1.shifted_shuffle(p2)) # needs sage.combinat sage.graphs sage.modules sage.rings.finite_rings ....: == sorted([Permutation(p) for p in ....: Word(p1).shifted_shuffle(Word(p2))]) ....: for p2 in Permutations(3) ) ....: for p1 in Permutations(2) ) True
>>> from sage.all import * >>> all( all( sorted(p1.shifted_shuffle(p2)) # needs sage.combinat sage.graphs sage.modules sage.rings.finite_rings ... == sorted([Permutation(p) for p in ... Word(p1).shifted_shuffle(Word(p2))]) ... for p2 in Permutations(Integer(3)) ) ... for p1 in Permutations(Integer(2)) ) True
- show(representation='cycles', orientation='landscape', **args)[source]¶
Display the permutation as a drawing.
INPUT:
representation
– different kinds of drawings are available'cycles'
– default; the permutation is displayed as a collection of directed cycles'braid'
– the permutation is displayed as segments linking each element \(1, ..., n\) to its image on a parallel lineWhen using this drawing, it is also possible to display the permutation horizontally (
orientation = "landscape"
, default option) or vertically (orientation = "portrait"
).'chord-diagram'
– the permutation is displayed as a directed graph, all of its vertices being located on a circle
All additional arguments are forwarded to the
show
subcalls.EXAMPLES:
sage: P20 = Permutations(20) sage: P20.random_element().show(representation='cycles') # needs sage.graphs sage.plot sage: P20.random_element().show(representation='chord-diagram') # needs sage.graphs sage.plot sage: P20.random_element().show(representation='braid') # needs sage.plot sage: P20.random_element().show(representation='braid', # needs sage.plot ....: orientation='portrait')
>>> from sage.all import * >>> P20 = Permutations(Integer(20)) >>> P20.random_element().show(representation='cycles') # needs sage.graphs sage.plot >>> P20.random_element().show(representation='chord-diagram') # needs sage.graphs sage.plot >>> P20.random_element().show(representation='braid') # needs sage.plot >>> P20.random_element().show(representation='braid', # needs sage.plot ... orientation='portrait')
- sign()[source]¶
Return the signature of the permutation
self
. This is \((-1)^l\), where \(l\) is the number of inversions ofself
.Note
sign()
can be used as an alias forsignature()
.EXAMPLES:
sage: Permutation([4, 2, 3, 1, 5]).signature() -1 sage: Permutation([1,3,2,5,4]).sign() 1 sage: Permutation([]).sign() 1
>>> from sage.all import * >>> Permutation([Integer(4), Integer(2), Integer(3), Integer(1), Integer(5)]).signature() -1 >>> Permutation([Integer(1),Integer(3),Integer(2),Integer(5),Integer(4)]).sign() 1 >>> Permutation([]).sign() 1
- signature()[source]¶
Return the signature of the permutation
self
. This is \((-1)^l\), where \(l\) is the number of inversions ofself
.Note
sign()
can be used as an alias forsignature()
.EXAMPLES:
sage: Permutation([4, 2, 3, 1, 5]).signature() -1 sage: Permutation([1,3,2,5,4]).sign() 1 sage: Permutation([]).sign() 1
>>> from sage.all import * >>> Permutation([Integer(4), Integer(2), Integer(3), Integer(1), Integer(5)]).signature() -1 >>> Permutation([Integer(1),Integer(3),Integer(2),Integer(5),Integer(4)]).sign() 1 >>> Permutation([]).sign() 1
- simion_schmidt(avoid=[1, 2, 3])[source]¶
Implement the Simion-Schmidt map which sends an arbitrary permutation to a pattern avoiding permutation, where the permutation pattern is one of four length-three patterns. This method also implements the bijection between (for example)
[1,2,3]
- and[1,3,2]
-avoiding permutations.INPUT:
avoid
– one of the patterns[1,2,3]
,[1,3,2]
,[3,1,2]
,[3,2,1]
EXAMPLES:
sage: P = Permutations(6) sage: p = P([4,5,1,6,3,2]) sage: pl = [ [1,2,3], [1,3,2], [3,1,2], [3,2,1] ] sage: for q in pl: # needs sage.combinat ....: s = p.simion_schmidt(q) ....: print("{} {}".format(s, s.has_pattern(q))) [4, 6, 1, 5, 3, 2] False [4, 2, 1, 3, 5, 6] False [4, 5, 3, 6, 2, 1] False [4, 5, 1, 6, 2, 3] False
>>> from sage.all import * >>> P = Permutations(Integer(6)) >>> p = P([Integer(4),Integer(5),Integer(1),Integer(6),Integer(3),Integer(2)]) >>> pl = [ [Integer(1),Integer(2),Integer(3)], [Integer(1),Integer(3),Integer(2)], [Integer(3),Integer(1),Integer(2)], [Integer(3),Integer(2),Integer(1)] ] >>> for q in pl: # needs sage.combinat ... s = p.simion_schmidt(q) ... print("{} {}".format(s, s.has_pattern(q))) [4, 6, 1, 5, 3, 2] False [4, 2, 1, 3, 5, 6] False [4, 5, 3, 6, 2, 1] False [4, 5, 1, 6, 2, 3] False
- size()[source]¶
Return the size of
self
.EXAMPLES:
sage: Permutation([3,4,1,2,5]).size() 5
>>> from sage.all import * >>> Permutation([Integer(3),Integer(4),Integer(1),Integer(2),Integer(5)]).size() 5
- stack_sort()[source]¶
Return the stack sort of a permutation.
This is another permutation obtained through the process of sorting using one stack. If the result is the identity permutation, the original permutation is stack-sortable.
See Wikipedia article Stack-sortable_permutation
EXAMPLES:
sage: p = Permutation([2,1,5,3,4,9,7,8,6]) sage: p.stack_sort() [1, 2, 3, 4, 5, 7, 6, 8, 9] sage: S5 = Permutations(5) sage: len([1 for s in S5 if s.stack_sort() == S5.one()]) 42
>>> from sage.all import * >>> p = Permutation([Integer(2),Integer(1),Integer(5),Integer(3),Integer(4),Integer(9),Integer(7),Integer(8),Integer(6)]) >>> p.stack_sort() [1, 2, 3, 4, 5, 7, 6, 8, 9] >>> S5 = Permutations(Integer(5)) >>> len([Integer(1) for s in S5 if s.stack_sort() == S5.one()]) 42
- sylvester_class(left_to_right=False)[source]¶
Iterate over the equivalence class of the permutation
self
under sylvester congruence.Sylvester congruence is an equivalence relation on the set \(S_n\) of all permutations of \(n\). It is defined as the smallest equivalence relation such that every permutation of the form \(uacvbw\) with \(u\), \(v\) and \(w\) being words and \(a\), \(b\) and \(c\) being letters satisfying \(a \leq b < c\) is equivalent to the permutation \(ucavbw\). (Here, permutations are regarded as words by way of one-line notation.) This definition comes from [HNT2005], Definition 8, where it is more generally applied to arbitrary words.
The equivalence class of a permutation \(p \in S_n\) under sylvester congruence is called the sylvester class of \(p\). It is an interval in the right permutohedron order (see
permutohedron_lequal()
) on \(S_n\).This is related to the
sylvester_class()
method in that the equivalence class of a permutation \(\pi\) under sylvester congruence is the sylvester class of the right-to-left binary search tree of \(\pi\). However, the present method yields permutations, while the method on labelled binary trees yields plain lists.If the variable
left_to_right
is set toTrue
, the method instead iterates over the equivalence class ofself
with respect to the left sylvester congruence. The left sylvester congruence is easiest to define by saying that two permutations are equivalent under it if and only if their reverses (reverse()
) are equivalent under (standard) sylvester congruence.EXAMPLES:
The sylvester class of a permutation in \(S_5\):
sage: p = Permutation([3, 5, 1, 2, 4]) sage: sorted(p.sylvester_class()) # needs sage.combinat sage.graphs [[1, 3, 2, 5, 4], [1, 3, 5, 2, 4], [1, 5, 3, 2, 4], [3, 1, 2, 5, 4], [3, 1, 5, 2, 4], [3, 5, 1, 2, 4], [5, 1, 3, 2, 4], [5, 3, 1, 2, 4]]
>>> from sage.all import * >>> p = Permutation([Integer(3), Integer(5), Integer(1), Integer(2), Integer(4)]) >>> sorted(p.sylvester_class()) # needs sage.combinat sage.graphs [[1, 3, 2, 5, 4], [1, 3, 5, 2, 4], [1, 5, 3, 2, 4], [3, 1, 2, 5, 4], [3, 1, 5, 2, 4], [3, 5, 1, 2, 4], [5, 1, 3, 2, 4], [5, 3, 1, 2, 4]]
The sylvester class of a permutation \(p\) contains \(p\):
sage: all(p in p.sylvester_class() for p in Permutations(4)) # needs sage.combinat sage.graphs True
>>> from sage.all import * >>> all(p in p.sylvester_class() for p in Permutations(Integer(4))) # needs sage.combinat sage.graphs True
Small cases:
sage: list(Permutation([]).sylvester_class()) # needs sage.combinat sage.graphs [[]] sage: list(Permutation([1]).sylvester_class()) # needs sage.combinat sage.graphs [[1]]
>>> from sage.all import * >>> list(Permutation([]).sylvester_class()) # needs sage.combinat sage.graphs [[]] >>> list(Permutation([Integer(1)]).sylvester_class()) # needs sage.combinat sage.graphs [[1]]
The sylvester classes in \(S_3\):
sage: [sorted(p.sylvester_class()) for p in Permutations(3)] # needs sage.combinat sage.graphs [[[1, 2, 3]], [[1, 3, 2], [3, 1, 2]], [[2, 1, 3]], [[2, 3, 1]], [[1, 3, 2], [3, 1, 2]], [[3, 2, 1]]]
>>> from sage.all import * >>> [sorted(p.sylvester_class()) for p in Permutations(Integer(3))] # needs sage.combinat sage.graphs [[[1, 2, 3]], [[1, 3, 2], [3, 1, 2]], [[2, 1, 3]], [[2, 3, 1]], [[1, 3, 2], [3, 1, 2]], [[3, 2, 1]]]
The left sylvester classes in \(S_3\):
sage: [sorted(p.sylvester_class(left_to_right=True)) # needs sage.combinat sage.graphs ....: for p in Permutations(3)] [[[1, 2, 3]], [[1, 3, 2]], [[2, 1, 3], [2, 3, 1]], [[2, 1, 3], [2, 3, 1]], [[3, 1, 2]], [[3, 2, 1]]]
>>> from sage.all import * >>> [sorted(p.sylvester_class(left_to_right=True)) # needs sage.combinat sage.graphs ... for p in Permutations(Integer(3))] [[[1, 2, 3]], [[1, 3, 2]], [[2, 1, 3], [2, 3, 1]], [[2, 1, 3], [2, 3, 1]], [[3, 1, 2]], [[3, 2, 1]]]
A left sylvester class in \(S_5\):
sage: p = Permutation([4, 2, 1, 5, 3]) sage: sorted(p.sylvester_class(left_to_right=True)) # needs sage.combinat sage.graphs [[4, 2, 1, 3, 5], [4, 2, 1, 5, 3], [4, 2, 3, 1, 5], [4, 2, 3, 5, 1], [4, 2, 5, 1, 3], [4, 2, 5, 3, 1], [4, 5, 2, 1, 3], [4, 5, 2, 3, 1]]
>>> from sage.all import * >>> p = Permutation([Integer(4), Integer(2), Integer(1), Integer(5), Integer(3)]) >>> sorted(p.sylvester_class(left_to_right=True)) # needs sage.combinat sage.graphs [[4, 2, 1, 3, 5], [4, 2, 1, 5, 3], [4, 2, 3, 1, 5], [4, 2, 3, 5, 1], [4, 2, 5, 1, 3], [4, 2, 5, 3, 1], [4, 5, 2, 1, 3], [4, 5, 2, 3, 1]]
- to_alternating_sign_matrix()[source]¶
Return a matrix representing the permutation in the
AlternatingSignMatrix
class.EXAMPLES:
sage: m = Permutation([1,2,3]).to_alternating_sign_matrix(); m # needs sage.combinat sage.modules [1 0 0] [0 1 0] [0 0 1] sage: parent(m) # needs sage.combinat sage.modules Alternating sign matrices of size 3
>>> from sage.all import * >>> m = Permutation([Integer(1),Integer(2),Integer(3)]).to_alternating_sign_matrix(); m # needs sage.combinat sage.modules [1 0 0] [0 1 0] [0 0 1] >>> parent(m) # needs sage.combinat sage.modules Alternating sign matrices of size 3
- to_cycles(singletons=True, use_min=True)[source]¶
Return the permutation
self
as a list of disjoint cycles.The cycles are returned in the order of increasing smallest elements, and each cycle is returned as a tuple which starts with its smallest element.
If
singletons=False
is given, the list does not contain the singleton cycles.If
use_min=False
is given, the cycles are returned in the order of increasing largest (not smallest) elements, and each cycle starts with its largest element.EXAMPLES:
sage: Permutation([2,1,3,4]).to_cycles() [(1, 2), (3,), (4,)] sage: Permutation([2,1,3,4]).to_cycles(singletons=False) [(1, 2)] sage: Permutation([2,1,3,4]).to_cycles(use_min=True) [(1, 2), (3,), (4,)] sage: Permutation([2,1,3,4]).to_cycles(use_min=False) [(4,), (3,), (2, 1)] sage: Permutation([2,1,3,4]).to_cycles(singletons=False, use_min=False) [(2, 1)] sage: Permutation([4,1,5,2,6,3]).to_cycles() [(1, 4, 2), (3, 5, 6)] sage: Permutation([4,1,5,2,6,3]).to_cycles(use_min=False) [(6, 3, 5), (4, 2, 1)] sage: Permutation([6, 4, 5, 2, 3, 1]).to_cycles() [(1, 6), (2, 4), (3, 5)] sage: Permutation([6, 4, 5, 2, 3, 1]).to_cycles(use_min=False) [(6, 1), (5, 3), (4, 2)]
>>> from sage.all import * >>> Permutation([Integer(2),Integer(1),Integer(3),Integer(4)]).to_cycles() [(1, 2), (3,), (4,)] >>> Permutation([Integer(2),Integer(1),Integer(3),Integer(4)]).to_cycles(singletons=False) [(1, 2)] >>> Permutation([Integer(2),Integer(1),Integer(3),Integer(4)]).to_cycles(use_min=True) [(1, 2), (3,), (4,)] >>> Permutation([Integer(2),Integer(1),Integer(3),Integer(4)]).to_cycles(use_min=False) [(4,), (3,), (2, 1)] >>> Permutation([Integer(2),Integer(1),Integer(3),Integer(4)]).to_cycles(singletons=False, use_min=False) [(2, 1)] >>> Permutation([Integer(4),Integer(1),Integer(5),Integer(2),Integer(6),Integer(3)]).to_cycles() [(1, 4, 2), (3, 5, 6)] >>> Permutation([Integer(4),Integer(1),Integer(5),Integer(2),Integer(6),Integer(3)]).to_cycles(use_min=False) [(6, 3, 5), (4, 2, 1)] >>> Permutation([Integer(6), Integer(4), Integer(5), Integer(2), Integer(3), Integer(1)]).to_cycles() [(1, 6), (2, 4), (3, 5)] >>> Permutation([Integer(6), Integer(4), Integer(5), Integer(2), Integer(3), Integer(1)]).to_cycles(use_min=False) [(6, 1), (5, 3), (4, 2)]
The algorithm is of complexity \(O(n)\) where \(n\) is the size of the given permutation.