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
trac ticket #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
Returns the product of 

Returns the product of 

Returns the size of the permutation 

Returns the disjointcycles representation of 

Returns the permutation that follows 

Returns the permutation that comes directly before 

Returns a tableau of shape 

Returns the permutation 

Return 

Returns a 

Returns the signature of the permutation 

Returns 

Returns a matrix representing the permutation 

Returns the rank of 

Returns the inversion vector of a permutation 

Returns a list of the inversions of permutation 

Returns the permutation obtained by sorting 

Return a digraph representation of 

Displays the permutation as a drawing. 

Returns the number of inversions in the permutation 

Returns the 

Returns the number of 

Returns the Coxeter length of a permutation 

Returns the inverse of a permutation 

Returns the 

Returns the 

Returns a list of the runs in the permutation 

Returns the length of the longest increasing subsequences of 

Returns the list of the longest increasing subsequences of 

Returns the cycle type of 

Returns the image of the permutation 

Returns the image of the permutation 

Returns the image of the permutation 

Returns the image of the permutation 

Return destandardization of 

Returns the Lehmer code of the permutation 

Returns the Lehmer cocode of 

Returns the reduced word of the permutation 

Returns a list of the reduced words of the permutation 

An iterator for the reduced words of the permutation 

Returns a lexicographically minimal reduced word of a permutation 

Returns a list of the fixed points of the permutation 

Returns the number of fixed points of the permutation 

Returns the list of the positions of the recoils of the permutation 

Returns the number of recoils of the permutation 

Returns the composition corresponding to the recoils of 

Returns the list of the descents of the permutation 

Returns a list of the idescents of 

Returns the list obtained by mapping each position in 

Returns the number of descents of the permutation 

Returns the number of idescents of the permutation 

Returns the composition corresponding to the descents of 

Returns the descent polynomial of the permutation 

Returns the major index of the permutation 

Returns the inverse major index of the permutation 

Returns the major code of the permutation 

Returns a list of the peaks of the permutation 

Returns the number of peaks of the permutation 

Returns a list of the saliances of the permutation 

Returns the number of saliances of the permutation 

Returns 

Returns all the numbers 

Returns the list of inversions of 

Returns an iterator over Bruhat inversions of 

Returns a list of the permutations covering 

An iterator for the permutations covering 

Returns a list of the permutations covered by 

An iterator for the permutations covered by 

Returns the combinatorial class of permutations smaller than or equal to 

Returns the combinatorial class of permutations greater than or equal to 

Returns 

Returns a list of the permutations covering 

Returns a list of the permutations covered by 

Returns a list of permutations smaller than or equal to 

Returns a list of permutations greater than or equal to 

Returns an iterator over permutations in an interval of the permutohedron order. 

Returns a list of permutations in an interval of the permutohedron order. 

Tests whether the permutation 

Tests whether the permutation 

Returns the list of positions where the pattern 

Returns the permutation obtained by reversing the 1line notation of 

Returns the complement of the permutation which is obtained by replacing each value \(x\) in the 1line notation of 

Returns the permutation poset of 

Returns a dictionary corresponding to the permutation 

Returns the action of the permutation 

Returns the pair of standard tableaux obtained by running the RobinsonSchensted Algorithm on 

Returns the left standard tableau after performing the RSK algorithm. 

Returns the right standard tableau after performing the RSK algorithm. 

Returns the increasing tree of 

Returns the shape of the increasing tree of 

Returns the binary search tree of 

Iterates over the equivalence class of 

Returns the shape of the tableaux obtained by the RSK algorithm. 

Returns the permutation obtained by removing any fixed points at the end of 

Returns the plain retract of 

Returns the directproduct retract of 

Returns the OkounkovVershik retract of 

Returns the cosettype of 

Returns the shape of the binary search tree of 

Returns the right (or left) shifted concatenation of 

Returns the shifted shuffle of 
Other classes defined in this file
Functions defined in this file
Returns the permutation corresponding to major code 

Returns a Permutation give a 

Returns the permutation with the specified lexicographic rank. 

Returns the permutation corresponding to inversion vector 

Returns the permutation with given disjointcycle representation 

Returns the permutation with Lehmer code 

Returns the permutation corresponding to the reduced word 

Returns a given bistochastic matrix as a nonnegative linear combination of permutations. 

Returns a partial permutation representing the bounded affine permutation of a matrix. 

Returns a list of all the permutations in a given descent class (i. e., having a given descents composition). 

Returns the smallest element of a descent class. 

Returns the largest element of a descent class. 

Returns 

Returns 

Returns a standard permutation corresponding to the permutation 
AUTHORS:
Mike Hansen
Dan Drake (20080407): allow Permutation() to take lists of tuples
Sébastien Labbé (20090317): added robinson_schensted_inverse
Travis Scrimshaw:
(20120816):
to_standard()
no longer modifies input(20130119): Removed RSK implementation and moved to
rsk
.(20130713): Removed
CombinatorialClass
and moved permutations to the category framework.
Darij Grinberg (20130907): added methods; ameliorated trac ticket #14885 by exposing and documenting methods for globalindependent multiplication.
Travis Scrimshaw (20140205): Made
StandardPermutations_n
a finite Weyl group to make it more uniform withSymmetricGroup
. Added ability to compute the conjugacy classes.
Classes and methods¶
 class sage.combinat.permutation.Arrangements¶
Bases:
sage.combinat.permutation.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() [[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() 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']]
 cardinality()¶
Return the cardinality of
self
.EXAMPLES:
sage: A = Arrangements([1,1,2,3,4,4,5], 2) sage: A.cardinality() 22
 class sage.combinat.permutation.Arrangements_msetk(mset, k)¶
Bases:
sage.combinat.permutation.Arrangements
,sage.combinat.permutation.Permutations_msetk
Arrangements of length \(k\) of a multiset \(M\).
 class sage.combinat.permutation.Arrangements_setk(s, k)¶
Bases:
sage.combinat.permutation.Arrangements
,sage.combinat.permutation.Permutations_setk
Arrangements of length \(k\) of a set \(S\).
 class sage.combinat.permutation.CyclicPermutations(mset)¶
Bases:
sage.combinat.permutation.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() [[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() [[1, 1, 1]]
 iterator(distinct=False)¶
EXAMPLES:
sage: CyclicPermutations(range(4)).list() # indirect doctest [[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() [[1, 1, 1]] sage: CyclicPermutations([1,1,1]).list(distinct=True) [[1, 1, 1], [1, 1, 1]]
 list(distinct=False)¶
EXAMPLES:
sage: CyclicPermutations(range(4)).list() [[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)¶
Bases:
sage.combinat.permutation.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() [[[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() [[[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() [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]
sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list(distinct=True) [[[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¶
Bases:
sage.structure.list_clone.ClonableArray
A cyclic permutation of a partition.
 check()¶
Check that
self
is a valid element.EXAMPLES:
sage: CP = CyclicPermutationsOfPartition([[1,2,3,4],[5,6,7]]) sage: elt = CP[0] sage: elt.check()
 iterator(distinct=False)¶
AUTHORS:
Robert Miller
EXAMPLES:
sage: CyclicPermutationsOfPartition([[1,2,3,4],[5,6,7]]).list() # indirect doctest [[[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() [[[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() [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]
sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list(distinct=True) [[[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)¶
EXAMPLES:
sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list() [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]] sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list(distinct=True) [[[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)¶
Bases:
sage.combinat.backtrack.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...>
 class sage.combinat.permutation.Permutation(parent, l, check_input=True)¶
Bases:
sage.combinat.combinat.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 oneline permutation notation. The construction checks that you give an acceptable entry. To avoid the check, use the
check_input
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 RobinsonSchensted algorithm.
check_input
(boolean) – whether to check that input is correct. Slowsthe function down, but ensures that nothing bad happens. This is set to
True
by default.
Warning
Since trac ticket #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_input = 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 trac ticket #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'>
Construction from a string in cycle notation:
sage: p = Permutation( '(4,5)' ); p [1, 2, 3, 5, 4]
The size of the permutation is the maximum integer appearing; add a 1cycle 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
We construct a
Permutation
from aPermutationGroupElement
:sage: g = PermutationGroupElement([2,1,3]) sage: Permutation(g) [2, 1, 3]
From a pair of tableaux of the same shape. This uses the inverse of the RobinsonSchensted algorithm:
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]
 RS_partition()¶
Return the shape of the tableaux obtained by applying the RSK algorithm to
self
.EXAMPLES:
sage: Permutation([1,4,3,2]).RS_partition() [2, 1, 1]
 absolute_length()¶
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() 1
 action(a)¶
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]
 avoids(patt)¶
Test whether the permutation
self
avoids the patternpatt
.EXAMPLES:
sage: Permutation([6,2,5,4,3,1]).avoids([4,2,3,1]) False sage: Permutation([6,1,2,5,4,3]).avoids([4,2,3,1]) True sage: Permutation([6,1,2,5,4,3]).avoids([3,4,1,2]) True
 binary_search_tree(left_to_right=True)¶
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 oneline 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() 1[., 4[3[2[., .], .], .]] sage: Permutation([4,1,3,2]).binary_search_tree() 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) 2[1[., .], 3[., 4[., .]]] sage: Permutation([4,1,3,2]).binary_search_tree(False) 2[1[., .], 3[., 4[., .]]]
 binary_search_tree_shape(left_to_right=True)¶
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() [., [[[., .], .], .]] sage: Permutation([4,1,3,2]).binary_search_tree_shape() [[., [[., .], .]], .]
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) [[., .], [., [., .]]] sage: Permutation([4,1,3,2]).binary_search_tree_shape(False) [[., .], [., [., .]]]
 bruhat_greater()¶
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]]
 bruhat_inversions()¶
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]]
 bruhat_inversions_iterator()¶
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]]
 bruhat_lequal(p2)¶
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
 bruhat_pred()¶
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]]
 bruhat_pred_iterator()¶
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]]
 bruhat_smaller()¶
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]]
 bruhat_succ()¶
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]]
 bruhat_succ_iterator()¶
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]]
 complement()¶
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]
 cycle_string(singletons=False)¶
Return a string of the permutation in cycle notation.
If
singletons=True
, it includes 1cycles 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)'
 cycle_tuples(singletons=True, use_min=True)¶
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)]
The algorithm is of complexity \(O(n)\) where \(n\) is the size of the given permutation.
 cycle_type()¶
Return a partition of
len(self)
corresponding to the cycle type ofself
.This is a nonincreasing sequence of the cycle lengths of
self
.EXAMPLES:
sage: Permutation([3,1,2,4]).cycle_type() [3, 1]
 decreasing_runs(as_tuple=False)¶
Decreasing runs of the permutation.
INPUT:
as_tuple
– boolean (default:False
) choice of output format
OUTPUT:
a 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,))
 descent_polynomial()¶
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() z1 sage: Permutation([4,3,2,1]).descent_polynomial() z1*z2^2*z3^3
Todo
This docstring needs to be fixed. First, the definition does not match the implementation (or the examples). Second, this doesn’t seem to be 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)¶
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 (defaultFalse
); ifTrue
, the last position of a nonempty permutation is also considered as a descentside
–'right'
(default) or'left'
; if'left'
, return the descents of the inverse permutationpositive
– boolean (defaultFalse
); ifTrue
, return the positions that are not descentsfrom_zero
– boolean (defaultFalse
); ifTrue
, return the positions starting from \(0\)index_set
– list (default:[1, ..., n1]
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]
 descents_composition()¶
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 \(i \in \{ 1, 2, \ldots, n1 \}\) 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 \(\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k1} \}\).
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() []
 destandardize(weight, ordered_alphabet=None)¶
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]) 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!
 dict()¶
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
 fixed_points()¶
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]
 foata_bijection()¶
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
\(1\),
\(14 \to 14\),
\(142 \to 412\),
\(4125 \to 4125\),
\(41253 \to 45123\).
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
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]
Border cases:
sage: Permutation([]).foata_bijection() [] sage: Permutation([1]).foata_bijection() [1]
 foata_bijection_inverse()¶
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]
 forget_cycles()¶
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 1line 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]
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]]
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)]) sage: R.<q> = QQ[] sage: sum(q^B.rank_function()(x) for x in B) 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)]) sage: B.is_isomorphic(B2) True
See also
 fundamental_transformation()¶
Return the image of the permutation
self
under the RenyiFoataSchuetzenberger 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 oneline 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 nonbijective) 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]
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]
 fundamental_transformation_inverse()¶
Return the image of the permutation
self
under the inverse of the RenyiFoataSchuetzenberger 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 lefttoright maxima of \(w\) (that is, of all indices \(j\) such that \(w(j)\) is bigger than each of \(w(1), w(2), \ldots, w(j1)\)). The image of \(w\) under the inverse of the fundamental transformation is the permutation \(u\) that sends \(w(i1)\) 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 \(w(i_{p1})\). 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]
 grade()¶
Return the size of
self
.EXAMPLES:
sage: Permutation([3,4,1,2,5]).size() 5
 has_pattern(patt)¶
Test whether the permutation
self
contains the patternpatt
.EXAMPLES:
sage: Permutation([3,5,1,4,6,2]).has_pattern([1,3,2]) True
 hyperoctahedral_double_coset_type()¶
Return the cosettype of
self
as a partition.self
must be a permutation of even size \(2n\). The cosettype 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 cosettype is determined as follows. Consider the perfect matching \(\{\{1,2\},\{3,4\},\dots,\{2n1,2n\}\}\) and its image by
self
, and draw them simultaneously as edges of a graph whose vertices are labeled by \(1,2,\dots,2n\). The cosettype is the ordered sequence of the semilengths of the cycles of this graph (see Chapter VII of [Mac1995] for more details, particularly Section VII.2).EXAMPLES:
sage: Permutation([3, 4, 6, 1, 5, 7, 2, 8]).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 cosettype
 idescents(final_descent=False, from_zero=False)¶
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 (defaultFalse
); ifTrue
, the last position of a nonempty permutation is also considered as a descentfrom_zero
– optional boolean (defaultFalse
); 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]
 idescents_signature(final_descent=False)¶
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 nonempty 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]
 imajor_index(final_descent=False)¶
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 0based, we need to add the number of descents.
With the
final_descent
option, the last position of a nonempty 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
 increasing_tree(compare=<builtin function min>)¶
Return the increasing tree associated to
self
.EXAMPLES:
sage: Permutation([1,4,3,2]).increasing_tree() 1[., 2[3[4[., .], .], .]] sage: Permutation([4,1,3,2]).increasing_tree() 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) 4[3[2[., .], .], 1[., .]] sage: Permutation([2,3,1,4]).increasing_tree(max) 4[3[2[., .], 1[., .]], .]
 increasing_tree_shape(compare=<builtin function min>)¶
Return the shape of the increasing tree associated with the permutation.
EXAMPLES:
sage: Permutation([1,4,3,2]).increasing_tree_shape() [., [[[., .], .], .]] sage: Permutation([4,1,3,2]).increasing_tree_shape() [[., .], [[., .], .]]
By passing the option
compare=max
one can have the decreasing tree instead:sage: Permutation([2,3,4,1]).increasing_tree_shape(max) [[[., .], .], [., .]] sage: Permutation([2,3,1,4]).increasing_tree_shape(max) [[[., .], [., .]], .]
 inverse()¶
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]
 inversions()¶
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)]
 is_derangement()¶
Return if
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
 is_even()¶
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
 ishift(i)¶
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 \(i1\) and \(i+1\). The \(i\)shift moves \(i\) to the other side, and leaves the relative positions of \(i1\) 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]
All entries other than \(i\), \(i1\) and \(i+1\) are unchanged:
sage: Permutation([2,4,1,3]).ishift(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]
 iswitch(i)¶
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 \(i1\), \(i\) 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]
All entries other than \(i1\), \(i\) and \(i+1\) are unchanged:
sage: Permutation([2,4,1,3]).iswitch(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]
 left_action_product(lp)¶
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]
 left_tableau()¶
Return the left standard tableau after performing the RSK algorithm on
self
.EXAMPLES:
sage: Permutation([1,4,3,2]).left_tableau() [[1, 2], [3], [4]]
 length()¶
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
 longest_increasing_subsequence_length()¶
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]) for i in Permutations(5)) True sage: Permutation([]).longest_increasing_subsequence_length() 0
 longest_increasing_subsequences()¶
Return the list of the longest increasing subsequences of
self
.Note
The algorithm is not optimal.
EXAMPLES:
sage: Permutation([2,3,4,1]).longest_increasing_subsequences() [[2, 3, 4]] sage: Permutation([5, 7, 1, 2, 6, 4, 3]).longest_increasing_subsequences() [[1, 2, 6], [1, 2, 4], [1, 2, 3]]
 major_index(final_descent=False)¶
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 0based, we need to add the number of descents.
With the
final_descent
option, the last position of a nonempty 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
 next()¶
Return the permutation that follows
self
in lexicographic order on the symmetric group containingself
. Ifself
is the last permutation, thennext
returnsFalse
.EXAMPLES:
sage: p = Permutation([1, 3, 2]) sage: next(p) [2, 1, 3] sage: p = Permutation([4,3,2,1]) sage: next(p) False
 noninversions(k)¶
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) []
 number_of_descents(final_descent=False)¶
Return the number of descents of
self
.With the
final_descent
option, the last position of a nonempty 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
 number_of_fixed_points()¶
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
 number_of_idescents(final_descent=False)¶
Return the number of idescents of
self
.See
idescents()
for a definition of idescents.With the
final_descent
option, the last position of a nonempty 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
 number_of_inversions()¶
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
 number_of_noninversions(k)¶
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
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) sage: all( x.number_of_noninversions(2) == b  x.number_of_inversions() ....: for x in Permutations(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
 number_of_peaks()¶
Return the number of peaks of the permutation
self
.A peak of a permutation \(p\) is an integer \(i\) such that \(p(i1) < p(i)\) 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
 number_of_recoils()¶
Return the number of recoils of the permutation
self
.EXAMPLES:
sage: Permutation([1,4,3,2]).number_of_recoils() 2
 number_of_saliances()¶
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
 pattern_positions(patt)¶
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]) [[0, 1, 3], [2, 3, 5], [2, 4, 5]]
 peaks()¶
Return a list of the peaks of the permutation
self
.A peak of a permutation \(p\) is an integer \(i\) such that \(p(i1) < p(i)\) 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() []
 permutation_poset()¶
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: 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]) sage: P.permutation_poset().greene_shape() == P.RS_partition() # This should hold for any P. True
 permutohedron_greater(side='right')¶
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]]
 permutohedron_join(other, side='right')¶
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 \(\{ 1, 2, ..., i1 \}\) 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 oneline 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]
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]
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
Border cases:
sage: p = Permutation([]) sage: p.permutohedron_join(p) [] sage: p = Permutation([1]) sage: 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]
 permutohedron_lequal(p2, side='right')¶
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
 permutohedron_meet(other, side='right')¶
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 \((n, n1, \ldots, 1) \in S_n\), 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]
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]
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
Border cases:
sage: p = Permutation([]) sage: p.permutohedron_meet(p) [] sage: p = Permutation([1]) sage: 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]
 permutohedron_pred(side='right')¶
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]]
 permutohedron_smaller(side='right')¶
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]]
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]]
 permutohedron_succ(side='right')¶
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]]
 prev()¶
Return the permutation that comes directly before
self
in lexicographic order on the symmetric group containingself
. Ifself
is the first permutation, then it returnsFalse
.EXAMPLES:
sage: p = Permutation([1,2,3]) sage: p.prev() False sage: p = Permutation([1,3,2]) sage: p.prev() [1, 2, 3]
 rank()¶
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
 recoils()¶
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() []
 recoils_composition()¶
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() []
 reduced_word()¶
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() []
 reduced_word_lexmin()¶
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() []
 reduced_words()¶
Return a list of the reduced words of
self
.The notion of a reduced word is based on the wellknown 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 \(i \in \{ 1, 2, \ldots, n1 \}\), 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 \(\{ 1, 2, \ldots, n1 \}\) (here \(\{ 1, 2, \ldots, n1 \}\) 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() [[]]
 reduced_words_iterator()¶
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]
 remove_extra_fixed_points()¶
Return the permutation obtained by removing any fixed points at the end of
self
.EXAMPLES:
sage: Permutation([2,1,3]).remove_extra_fixed_points() [2, 1] sage: Permutation([1,2,3,4]).remove_extra_fixed_points() [1]
See also
 retract_direct_product(m)¶
Return the directproduct 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 directproduct 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 oneline 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
See also
 retract_okounkov_vershik(m)¶
Return the OkounkovVershik 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 OkounkovVershik 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 OkounkovVershik 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 \(m = n1\), the OkounkovVershik retract (as a map \(S_n \to S_{n1}\)) is the map \(\widetilde{p}_n\) introduced in Section 7 of [VO2005], and appears as (3.20) in [CST2010]. In the general case, the OkounkovVershik retract of a permutation in \(S_n\) to \(S_m\) can be obtained by first taking its OkounkovVershik retract to \(S_{n1}\), then that of the resulting permutation to \(S_{n2}\), 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
See also
 retract_plain(m)¶
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 oneline 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
 reverse()¶
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]
 right_action_product(rp)¶
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]
 right_permutohedron_interval(other)¶
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: Permutation([2, 1, 4, 5, 3]).right_permutohedron_interval(Permutation([2, 5, 4, 1, 3])) [[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)¶
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: Permutation([2, 1, 4, 5, 3]).right_permutohedron_interval(Permutation([2, 5, 4, 1, 3])) # indirect doctest [[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()¶
Return the right standard tableau after performing the RSK algorithm on
self
.EXAMPLES:
sage: Permutation([1,4,3,2]).right_tableau() [[1, 2], [3], [4]]
 robinson_schensted()¶
Return the pair of standard tableaux obtained by running the RobinsonSchensted 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() [[[1, 3, 4], [2, 5], [6, 7]], [[1, 3, 5], [2, 6], [4, 7]]]
 runs(as_tuple=False)¶
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:
a 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]]
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,))
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
 saliances()¶
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]
 shifted_concatenation(other, side='right')¶
Return the right (or left) shifted concatenation of
self
with a permutationother
. These operations are also known as the LodayRonco over and under operations.INPUT:
other
– a permutation, a list, a tuple, or any iterable representing a permutation.side
– (default:"right"
) the string “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]
 shifted_shuffle(other)¶
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: 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: len(Permutation([3, 1, 5, 4, 2]).shifted_shuffle(Permutation([2, 1, 4, 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) ....: 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
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)) ....: == sorted([Permutation(p) for p in ....: Word(p1).shifted_shuffle(Word(p2))]) ....: for p2 in Permutations(3) ) ....: for p1 in Permutations(2) ) True
 show(representation='cycles', orientation='landscape', **args)¶
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 line.When using this drawing, it is also possible to display the permutation horizontally (
orientation = "landscape"
, default option) or vertically (orientation = "portrait"
)."chorddiagram"
– 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: Permutations(20).random_element().show(representation = "cycles") sage: Permutations(20).random_element().show(representation = "chorddiagram") sage: Permutations(20).random_element().show(representation = "braid") sage: Permutations(20).random_element().show(representation = "braid", orientation='portrait')
 sign()¶
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
 signature()¶
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
 simion_schmidt(avoid=[1, 2, 3])¶
Implements the SimionSchmidt map which sends an arbitrary permutation to a pattern avoiding permutation, where the permutation pattern is one of four lengththree 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: ....: 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()¶
Return the size of
self
.EXAMPLES:
sage: Permutation([3,4,1,2,5]).size() 5
 stack_sort()¶
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 stacksortable.
See Wikipedia article Stacksortable_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
 sylvester_class(left_to_right=False)¶
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 oneline 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 righttoleft 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()) [[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) ) True
Small cases:
sage: list(Permutation([]).sylvester_class()) [[]] sage: list(Permutation([1]).sylvester_class()) [[1]]
The sylvester classes in \(S_3\):
sage: [sorted(p.sylvester_class()) for p in Permutations(3)] [[[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)) 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]]]
A left sylvester class in \(S_5\):
sage: p = Permutation([4, 2, 1, 5, 3]) sage: sorted(p.sylvester_class(left_to_right=True)) [[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()¶
Return a matrix representing the permutation in the
AlternatingSignMatrix
class.EXAMPLES:
sage: m = Permutation([1,2,3]).to_alternating_sign_matrix(); m [1 0 0] [0 1 0] [0 0 1] sage: parent(m) Alternating sign matrices of size 3
 to_cycles(singletons=True, use_min=True)¶
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)]
The algorithm is of complexity \(O(n)\) where \(n\) is the size of the given permutation.
 to_digraph()¶
Return a digraph representation of
self
.EXAMPLES:
sage: d = Permutation([3, 1, 2]).to_digraph() sage: d.edges(labels=False) [(1, 3), (2, 1), (3, 2)] sage: P = Permutations(range(1, 10)) sage: d = Permutation(P.random_element()).to_digraph() sage: all(c.is_cycle() for c in d.strongly_connected_components_subgraphs()) True
 to_inversion_vector()¶
Return the inversion vector of
self
.The inversion vector of a permutation \(p \in S_n\) is defined as the vector \((v_1, v_2, \ldots, v_n)\), where \(v_i\) is the number of elements larger than \(i\) that appear to the left of \(i\) in the permutation \(p\).
The algorithm is of complexity \(O(n\log(n))\) where \(n\) is the size of the given permutation.
EXAMPLES:
sage: Permutation([5,9,1,8,2,6,4,7,3]).to_inversion_vector() [2, 3, 6, 4, 0, 2, 2, 1, 0] sage: Permutation([8,7,2,1,9,4,6,5,10,3]).to_inversion_vector() [3, 2, 7, 3, 4, 3, 1, 0, 0, 0] sage: Permutation([3,2,4,1,5]).to_inversion_vector() [3, 1, 0, 0, 0]
 to_lehmer_cocode()¶
Return the Lehmer cocode of the permutation
self
.The Lehmer cocode of a permutation \(p\) is defined as the list \((c_1, c_2, \ldots, c_n)\), where \(c_i\) is the number of \(j < i\) such that \(p(j) > p(i)\).
EXAMPLES:
sage: p = Permutation([2,1,3]) sage: p.to_lehmer_cocode() [0, 1, 0] sage: q = Permutation([3,1,2]) sage: q.to_lehmer_cocode() [0, 1, 1]
 to_lehmer_code()¶
Return the Lehmer code of the permutation
self
.The Lehmer code of a permutation \(p\) is defined as the list \([c[1],c[2],...,c[n]]\), where \(c[i]\) is the number of \(j>i\) such that \(p(j)<p(i)\).
EXAMPLES:
sage: p = Permutation([2,1,3]) sage: p.to_lehmer_code() [1, 0, 0] sage: q = Permutation([3,1,2]) sage: q.to_lehmer_code() [2, 0, 0] sage: Permutation([1]).to_lehmer_code() [0] sage: Permutation([]).to_lehmer_code() []
 to_major_code(final_descent=False)¶
Return the major code of the permutation
self
.The major code of a permutation \(p\) is defined as the sequence \((m_1m_2, m_2m_3, \ldots, m_n)\), where \(m_i\) is the major index of the permutation obtained by erasing all letters smaller than \(i\) from \(p\).
With the
final_descent
option, the last position of a nonempty permutation is also considered as a descent. This has an effect on the computation of major indices.REFERENCES:
Carlitz, L. qBernoulli and Eulerian Numbers. Trans. Amer. Math. Soc. 76 (1954) 332350. http://www.ams.org/journals/tran/195407602/S00029947195400605382/
Skandera, M. An Eulerian Partner for Inversions. Sém. Lothar. Combin. 46 (2001) B46d. http://www.lehigh.edu/~mas906/papers/partner.ps
EXAMPLES:
sage: Permutation([9,3,5,7,2,1,4,6,8]).to_major_code() [5, 0, 1, 0, 1, 2, 0, 1, 0] sage: Permutation([2,8,4,3,6,7,9,5,1]).to_major_code() [8, 3, 3, 1, 4, 0, 1, 0, 0]
 to_matrix()¶
Return a matrix representing the permutation.
EXAMPLES:
sage: Permutation([1,2,3]).to_matrix() [1 0 0] [0 1 0] [0 0 1]
Alternatively:
sage: matrix(Permutation([1,3,2])) [1 0 0] [0 0 1] [0 1 0]
Notice that matrix multiplication corresponds to permutation multiplication only when the permutation option mult=’r2l’
sage: Permutations.options.mult='r2l' sage: p = Permutation([2,1,3]) sage: q = Permutation([3,1,2]) sage: (p*q).to_matrix() [0 0 1] [0 1 0] [1 0 0] sage: p.to_matrix()*q.to_matrix() [0 0 1] [0 1 0] [1 0 0] sage: Permutations.options.mult='l2r' sage: (p*q).to_matrix() [1 0 0] [0 0 1] [0 1 0]
 to_permutation_group_element()¶
Return a PermutationGroupElement equal to
self
.EXAMPLES:
sage: Permutation([2,1,4,3]).to_permutation_group_element() (1,2)(3,4) sage: Permutation([1,2,3]).to_permutation_group_element() ()
 to_tableau_by_shape(shape)¶
Return a tableau of shape
shape
with the entries inself
. The tableau is such that the reading word (i. e., the word obtained by reading the tableau row by row, starting from the top row in English notation, with each row being read from left to right) isself
.EXAMPLES:
sage: Permutation([3,4,1,2,5]).to_tableau_by_shape([3,2]) [[1, 2, 5], [3, 4]] sage: Permutation([3,4,1,2,5]).to_tableau_by_shape([3,2]).reading_word_permutation() [3, 4, 1, 2, 5]
 weak_excedences()¶
Return all the numbers
self[i]
such thatself[i] >= i+1
.EXAMPLES:
sage: Permutation([1,4,3,2,5]).weak_excedences() [1, 4, 3, 5]
 class sage.combinat.permutation.Permutations¶
Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.parent.Parent
Permutations.
Permutations(n)
returns the class of permutations ofn
, ifn
is an integer, list, set, or string.Permutations(n, k)
returns the class of lengthk
partial permutations ofn
(wheren
is any of the above things);k
must be a nonnegative integer. A length\(k\) partial permutation of \(n\) is defined as a \(k\)tuple of pairwise distinct elements of \(\{ 1, 2, \ldots, n \}\).Valid keyword arguments are: ‘descents’, ‘bruhat_smaller’, ‘bruhat_greater’, ‘recoils_finer’, ‘recoils_fatter’, ‘recoils’, and ‘avoiding’. With the exception of ‘avoiding’, you cannot specify
n
ork
along with a keyword.Permutations(descents=(list,n))
returns the class of permutations of \(n\) with descents in the positions specified bylist
. This uses the slightly nonstandard convention that the images of \(1,2,...,n\) under the permutation are regarded as positions \(0,1,...,n1\), so for example the presence of \(1\) inlist
signifies that the permutations \(\pi\) should satisfy \(\pi(2) > \pi(3)\). Note thatlist
is supposed to be a list of positions of the descents, not the descents composition. It does not return the class of permutations with descents compositionlist
.Permutations(bruhat_smaller=p)
andPermutations(bruhat_greater=p)
return the class of permutations smallerorequal or greaterorequal, respectively, than the given permutationp
in the Bruhat order. (The Bruhat order is defined inbruhat_lequal()
. It is also referred to as the strong Bruhat order.)Permutations(recoils=p)
returns the class of permutations whose recoils composition isp
. Unlike thedescents=(list, n)
syntax, this actually takes a composition as input.Permutations(recoils_fatter=p)
andPermutations(recoils_finer=p)
return the class of permutations whose recoils composition is fatter or finer, respectively, than the given compositionp
.Permutations(n, avoiding=P)
returns the class of permutations ofn
avoidingP
. HereP
may be a single permutation or a list of permutations; the returned class will avoid all patterns inP
.EXAMPLES:
sage: p = Permutations(3); p Standard permutations of 3 sage: p.list() [[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]]
sage: p = Permutations(3, 2); p Permutations of {1,...,3} of length 2 sage: p.list() [[1, 2], [1, 3], [2, 1], [2, 3], [3, 1], [3, 2]]
sage: p = Permutations(['c', 'a', 't']); p Permutations of the set ['c', 'a', 't'] sage: p.list() [['c', 'a', 't'], ['c', 't', 'a'], ['a', 'c', 't'], ['a', 't', 'c'], ['t', 'c', 'a'], ['t', 'a', 'c']]
sage: p = Permutations(['c', 'a', 't'], 2); p Permutations of the set ['c', 'a', 't'] of length 2 sage: p.list() [['c', 'a'], ['c', 't'], ['a', 'c'], ['a', 't'], ['t', 'c'], ['t', 'a']]
sage: p = Permutations([1,1,2]); p Permutations of the multiset [1, 1, 2] sage: p.list() [[1, 1, 2], [1, 2, 1], [2, 1, 1]]
sage: p = Permutations([1,1,2], 2); p Permutations of the multiset [1, 1, 2] of length 2 sage: p.list() [[1, 1], [1, 2], [2, 1]]
sage: p = Permutations(descents=([1], 4)); p Standard permutations of 4 with descents [1] sage: p.list() [[2, 4, 1, 3], [3, 4, 1, 2], [1, 4, 2, 3], [1, 3, 2, 4], [2, 3, 1, 4]]
sage: p = Permutations(bruhat_smaller=[1,3,2,4]); p Standard permutations that are less than or equal to [1, 3, 2, 4] in the Bruhat order sage: p.list() [[1, 2, 3, 4], [1, 3, 2, 4]]
sage: p = Permutations(bruhat_greater=[4,2,3,1]); p Standard permutations that are greater than or equal to [4, 2, 3, 1] in the Bruhat order sage: p.list() [[4, 2, 3, 1], [4, 3, 2, 1]]
sage: p = Permutations(recoils_finer=[2,1]); p Standard permutations whose recoils composition is finer than [2, 1] sage: p.list() [[3, 1, 2], [1, 2, 3], [1, 3, 2]]
sage: p = Permutations(recoils_fatter=[2,1]); p Standard permutations whose recoils composition is fatter than [2, 1] sage: p.list() [[3, 1, 2], [3, 2, 1], [1, 3, 2]]
sage: p = Permutations(recoils=[2,1]); p Standard permutations whose recoils composition is [2, 1] sage: p.list() [[3, 1, 2], [1, 3, 2]]
sage: p = Permutations(4, avoiding=[1,3,2]); p Standard permutations of 4 avoiding [[1, 3, 2]] sage: p.list() [[4, 1, 2, 3], [4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 1, 2], [4, 3, 2, 1], [3, 4, 1, 2], [3, 4, 2, 1], [2, 3, 4, 1], [3, 2, 4, 1], [1, 2, 3, 4], [2, 1, 3, 4], [2, 3, 1, 4], [3, 1, 2, 4], [3, 2, 1, 4]]
sage: p = Permutations(5, avoiding=[[3,4,1,2], [4,2,3,1]]); p Standard permutations of 5 avoiding [[3, 4, 1, 2], [4, 2, 3, 1]] sage: p.cardinality() 88 sage: p.random_element().parent() is p True
 Element¶
alias of
Permutation
 options(*get_value, **set_value)¶
Set the global options for elements of the permutation class. The defaults are for permutations to be displayed in list notation and the multiplication done from left to right (like in GAP) – that is, \((\pi \psi)(i) = \psi(\pi(i))\) for all \(i\).
Note
These options have no effect on permutation group elements.
OPTIONS:
display
– (default:list
) Specifies how the permutations should be printedcycle
– the permutations are displayed in cycle notation (i. e., as products of disjoint cycles)list
– the permutations are displayed in list notation (aka 1line notation)reduced_expression
– alias forreduced_word
reduced_word
– the permutations are displayed as reduced wordssingleton
– the permutations are displayed in cycle notation with singleton cycles shown as wellword
– alias forreduced_word
generator_name
– (default:s
) the letter used in latexing the reduced wordlatex
– (default:list
) Specifies how the permutations should be latexedcycle
– latex in cycle notationlist
– latex as a list in oneline notationoneline
– alias forlist
reduced_expression
– alias forreduced_word
reduced_word
– latex as reduced wordssingleton
– latex in cycle notation with singleton cycles shown as welltwoline
– latex in twoline notationword
– alias forreduced_word
latex_empty_str
– (default:1
) The LaTeX representation of a reduced word when said word is emptymult
– (default:l2r
) The multiplication of permutationsl2r
– left to right: \((p_1 \cdot p_2)(x) = p_2(p_1(x))\)r2l
– right to left: \((p_1 \cdot p_2)(x) = p_1(p_2(x))\)
EXAMPLES:
sage: p213 = Permutation([2,1,3]) sage: p312 = Permutation([3,1,2]) sage: Permutations.options(mult='l2r', display='list') sage: Permutations.options.display list sage: p213 [2, 1, 3] sage: Permutations.options.display='cycle' sage: p213 (1,2) sage: Permutations.options.display='singleton' sage: p213 (1,2)(3) sage: Permutations.options.display='list'
sage: Permutations.options.mult l2r sage: p213*p312 [1, 3, 2] sage: Permutations.options.mult='r2l' sage: p213*p312 [3, 2, 1] sage: Permutations.options._reset()
See
GlobalOptions
for more features of these options.
 class sage.combinat.permutation.PermutationsNK(s, k)¶
Bases:
sage.combinat.permutation.Permutations_setk
This exists solely for unpickling
PermutationsNK
objects created with Sage <= 6.3.
 class sage.combinat.permutation.Permutations_mset(mset)¶
Bases:
sage.combinat.permutation.Permutations
Permutations of a multiset \(M\).
A permutation of a multiset \(M\) is represented by a list that contains exactly the same elements as \(M\) (with the same multiplicities), but possibly in different order. If \(M\) is a proper set there are \(M !\) such permutations. Otherwise, if the first element appears \(k_1\) times, the second element appears \(k_2\) times and so on, the number of permutations is \(M! / (k_1! k_2! \ldots)\), which is sometimes called a multinomial coefficient.
EXAMPLES:
sage: mset = [1,1,2,2,2] sage: from sage.combinat.permutation import Permutations_mset sage: P = Permutations_mset(mset); P Permutations of the multiset [1, 1, 2, 2, 2] sage: sorted(P) [[1, 1, 2, 2, 2], [1, 2, 1, 2, 2], [1, 2, 2, 1, 2], [1, 2, 2, 2, 1], [2, 1, 1, 2, 2], [2, 1, 2, 1, 2], [2, 1, 2, 2, 1], [2, 2, 1, 1, 2], [2, 2, 1, 2, 1], [2, 2, 2, 1, 1]] sage: MS = MatrixSpace(GF(2),2,2) sage: A = MS([1,0,1,1]) sage: rows = A.rows() sage: rows[0].set_immutable() sage: rows[1].set_immutable() sage: P = Permutations_mset(rows); P Permutations of the multiset [(1, 0), (1, 1)] sage: sorted(P) [[(1, 0), (1, 1)], [(1, 1), (1, 0)]]
 class Element¶
Bases:
sage.structure.list_clone.ClonableArray
A permutation of an arbitrary multiset.
 check()¶
Verify that
self
is a valid permutation of the underlying multiset.EXAMPLES:
sage: S = Permutations(['c','a','c']) sage: elt = S(['c','c','a']) sage: elt.check()
 cardinality()¶
Return the cardinality of the set.
EXAMPLES:
sage: Permutations([1,2,2]).cardinality() 3 sage: Permutations([1,1,2,2,2]).cardinality() 10
 rank(p)¶
Return the rank of
p
in lexicographic order.INPUT:
p
– a permutation of \(M\)
ALGORITHM:
The algorithm uses the recurrence from the solution to exercise 4 in [Knu2011], Section 7.2.1.2:
\[\mathrm{rank}(p_1 \ldots p_n) = \mathrm{rank}(p_2 \ldots p_n) + \frac{1}{n} \genfrac{(}{)}{0pt}{0}{n}{n_1, \ldots, n_t} \sum_{j=1}^t n_j \left[ x_j < p_1 \right],\]where \(x_j, n_j\) are the distinct elements of \(p\) with their multiplicities, \(n\) is the sum of \(n_1, \ldots, n_t\), \(\genfrac{(}{)}{0pt}{1}{n}{n_1, \ldots, n_t}\) is the multinomial coefficient \(\frac{n!}{n_1! \ldots n_t!}\), and \(\sum_{j=1}^t n_j \left[ x_j < p_1 \right]\) means “the number of elements to the right of the first element that are less than the first element”.
EXAMPLES:
sage: mset = [1, 1, 2, 3, 4, 5, 5, 6, 9] sage: p = Permutations(mset) sage: p.rank(list(sorted(mset))) 0 sage: p.rank(list(reversed(sorted(mset)))) == p.cardinality()  1 True sage: p.rank([3, 1, 4, 1, 5, 9, 2, 6, 5]) 30991
 unrank(r)¶
Return the permutation of \(M\) having lexicographic rank
r
.INPUT:
r
– an integer between0
andself.cardinality()1
inclusive
ALGORITHM:
The algorithm is adapted from the solution to exercise 4 in [Knu2011], Section 7.2.1.2.
EXAMPLES:
sage: mset = [1, 1, 2, 3, 4, 5, 5, 6, 9] sage: p = Permutations(mset) sage: p.unrank(30991) [3, 1, 4, 1, 5, 9, 2, 6, 5] sage: p.rank(p.unrank(10)) 10 sage: p.unrank(0) == list(sorted(mset)) True sage: p.unrank(p.cardinality()1) == list(reversed(sorted(mset))) True
 class sage.combinat.permutation.Permutations_msetk(mset, k)¶
Bases:
sage.combinat.permutation.Permutations_mset
Length\(k\) partial permutations of a multiset.
A length\(k\) partial permutation of a multiset \(M\) is represented by a list of length \(k\) whose entries are elements of \(M\), appearing in the list with a multiplicity not higher than their respective multiplicity in \(M\).
 cardinality()¶
Return the cardinality of the set.
EXAMPLES:
sage: Permutations([1,2,2],2).cardinality() 3
 class sage.combinat.permutation.Permutations_nk(n, k)¶
Bases:
sage.combinat.permutation.Permutations
Length\(k\) partial permutations of \(\{1, 2, \ldots, n\}\).
 class Element¶
Bases:
sage.structure.list_clone.ClonableArray
A length\(k\) partial permutation of \([n]\).
 check()¶
Verify that
self
is a valid length\(k\) partial permutation of \([n]\).EXAMPLES:
sage: S = Permutations(4, 2) sage: elt = S([3, 1]) sage: elt.check()
 cardinality()¶
EXAMPLES:
sage: Permutations(3,0).cardinality() 1 sage: Permutations(3,1).cardinality() 3 sage: Permutations(3,2).cardinality() 6 sage: Permutations(3,3).cardinality() 6 sage: Permutations(3,4).cardinality() 0
 random_element()¶
EXAMPLES:
sage: s = Permutations(3,2).random_element() sage: s in Permutations(3,2) True
 class sage.combinat.permutation.Permutations_set(s)¶
Bases:
sage.combinat.permutation.Permutations
Permutations of an arbitrary given finite set.
Here, a “permutation of a finite set \(S\)” means a list of the elements of \(S\) in which every element of \(S\) occurs exactly once. This is not to be confused with bijections from \(S\) to \(S\), which are also often called permutations in literature.
 class Element¶
Bases:
sage.structure.list_clone.ClonableArray
A permutation of an arbitrary set.
 check()¶
Verify that
self
is a valid permutation of the underlying set.EXAMPLES:
sage: S = Permutations(['c','a','t']) sage: elt = S(['t','c','a']) sage: elt.check()
 cardinality()¶
Return the cardinality of the set.
EXAMPLES:
sage: Permutations([1,2,3]).cardinality() 6
 random_element()¶
EXAMPLES:
sage: s = Permutations([1,2,3]).random_element() sage: s.parent() is Permutations([1,2,3]) True
 class sage.combinat.permutation.Permutations_setk(s, k)¶
Bases:
sage.combinat.permutation.Permutations_set
Length\(k\) partial permutations of an arbitrary given finite set.
Here, a “length\(k\) partial permutation of a finite set \(S\)” means a list of length \(k\) whose entries are pairwise distinct and all belong to \(S\).
 random_element()¶
EXAMPLES:
sage: s = Permutations([1,2,4], 2).random_element() sage: s in Permutations([1,2,4], 2) True
 class sage.combinat.permutation.StandardPermutations_all¶
Bases:
sage.combinat.permutation.Permutations
All standard permutations.
 graded_component(n)¶
Return the graded component.
EXAMPLES:
sage: P = Permutations() sage: P.graded_component(4) == Permutations(4) True
 class sage.combinat.permutation.StandardPermutations_all_avoiding(a)¶
Bases:
sage.combinat.permutation.StandardPermutations_all
All standard permutations avoiding a set of patterns.
 patterns()¶
Return the patterns avoided by this class of permutations.
EXAMPLES:
sage: P = Permutations(avoiding=[[2,1,3],[1,2,3]]) sage: P.patterns() ([2, 1, 3], [1, 2, 3])
 class sage.combinat.permutation.StandardPermutations_avoiding_12(n)¶
Bases:
sage.combinat.permutation.StandardPermutations_avoiding_generic
 cardinality()¶
Return the cardinality of
self
.EXAMPLES:
sage: P = Permutations(3, avoiding=[1, 2]) sage: P.cardinality() 1
 class sage.combinat.permutation.StandardPermutations_avoiding_123(n)¶
Bases:
sage.combinat.permutation.StandardPermutations_avoiding_generic
 cardinality()¶
EXAMPLES:
sage: Permutations(5, avoiding=[1, 2, 3]).cardinality() 42 sage: len( Permutations(5, avoiding=[1, 2, 3]).list() ) 42
 class sage.combinat.permutation.StandardPermutations_avoiding_132(n)¶
Bases:
sage.combinat.permutation.StandardPermutations_avoiding_generic
 cardinality()¶
EXAMPLES:
sage: Permutations(5, avoiding=[1, 3, 2]).cardinality() 42 sage: len( Permutations(5, avoiding=[1, 3, 2]).list() ) 42
 class sage.combinat.permutation.StandardPermutations_avoiding_21(n)¶
Bases:
sage.combinat.permutation.StandardPermutations_avoiding_generic
 cardinality()¶
Return the cardinality of
self
.EXAMPLES:
sage: P = Permutations(3, avoiding=[2, 1]) sage: P.cardinality() 1
 class sage.combinat.permutation.StandardPermutations_avoiding_213(n)¶
Bases:
sage.combinat.permutation.StandardPermutations_avoiding_generic
 cardinality()¶
EXAMPLES:
sage: Permutations(5, avoiding=[2, 1, 3]).cardinality() 42 sage: len( Permutations(5, avoiding=[2, 1, 3]).list() ) 42
 class sage.combinat.permutation.StandardPermutations_avoiding_231(n)¶
Bases:
sage.combinat.permutation.StandardPermutations_avoiding_generic
 cardinality()¶
EXAMPLES:
sage: Permutations(5, avoiding=[2, 3, 1]).cardinality() 42 sage: len( Permutations(5, avoiding=[2, 3, 1]).list() ) 42
 class sage.combinat.permutation.StandardPermutations_avoiding_312(n)¶
Bases:
sage.combinat.permutation.StandardPermutations_avoiding_generic
 cardinality()¶
EXAMPLES:
sage: Permutations(5, avoiding=[3, 1, 2]).cardinality() 42 sage: len( Permutations(5, avoiding=[3, 1, 2]).list() ) 42
 class sage.combinat.permutation.StandardPermutations_avoiding_321(n)¶
Bases:
sage.combinat.permutation.StandardPermutations_avoiding_generic
 cardinality()¶
EXAMPLES:
sage: Permutations(5, avoiding=[3, 2, 1]).cardinality() 42 sage: len( Permutations(5, avoiding=[3, 2, 1]).list() ) 42
 class sage.combinat.permutation.StandardPermutations_avoiding_generic(n, a)¶
Bases:
sage.combinat.permutation.StandardPermutations_n_abstract
Generic class for subset of permutations avoiding a set of patterns.
 a¶
self.a
is deprecated; usepatterns()
instead.
 cardinality()¶
Return the cardinality of
self
.EXAMPLES:
sage: P = Permutations(3, avoiding=[[2, 1, 3],[1,2,3]]) sage: P.cardinality() 4
 patterns()¶
Return the patterns avoided by this class of permutations.
EXAMPLES:
sage: P = Permutations(3, avoiding=[[2,1,3],[1,2,3]]) sage: P.patterns() ([2, 1, 3], [1, 2, 3])
 class sage.combinat.permutation.StandardPermutations_bruhat_greater(p)¶
Bases:
sage.combinat.permutation.Permutations
Permutations of \(\{1, \ldots, n\}\) that are greater than or equal to a permutation \(p\) in the Bruhat order.
 class sage.combinat.permutation.StandardPermutations_bruhat_smaller(p)¶
Bases:
sage.combinat.permutation.Permutations
Permutations of \(\{1, \ldots, n\}\) that are less than or equal to a permutation \(p\) in the Bruhat order.
 class sage.combinat.permutation.StandardPermutations_descents(d, n)¶
Bases:
sage.combinat.permutation.StandardPermutations_n_abstract
Permutations of \(\{1, \ldots, n\}\) with a fixed set of descents.
 cardinality()¶
Return the cardinality of
self
.ALGORITHM:
The algorithm described in [Vie1979] is implemented naively.
EXAMPLES:
sage: P = Permutations(descents=([1,0,2], 5)) sage: P.cardinality() 4
 first()¶
Return the first permutation with descents \(d\).
EXAMPLES:
sage: Permutations(descents=([1,0,4,8],12)).first() [3, 2, 1, 4, 6, 5, 7, 8, 10, 9, 11, 12]
 last()¶
Return the last permutation with descents \(d\).
EXAMPLES:
sage: Permutations(descents=([1,0,4,8],12)).last() [12, 11, 8, 9, 10, 4, 5, 6, 7, 1, 2, 3]
 class sage.combinat.permutation.StandardPermutations_n(n)¶
Bases:
sage.combinat.permutation.StandardPermutations_n_abstract
Permutations of the set \(\{1, 2, \ldots, n\}\).
These are also called permutations of size \(n\), or the elements of the \(n\)th symmetric group.
Todo
Have a
reduced_word()
which works in both multiplication conventions. class Element(parent, l, check_input=True)¶
Bases:
sage.combinat.permutation.Permutation
 apply_simple_reflection_left(i)¶
Return
self
multiplied by the simple reflections[i]
on the left.This acts by switching the entries in positions \(i\) and \(i+1\).
Warning
This ignores the multiplication convention in order to be consistent with other Coxeter operations in permutations (e.g., computing
reduced_word()
).EXAMPLES:
sage: W = Permutations(3) sage: w = W([2,3,1]) sage: w.apply_simple_reflection_left(1) [1, 3, 2] sage: w.apply_simple_reflection_left(2) [3, 2, 1]
 apply_simple_reflection_right(i)¶
Return
self
multiplied by the simple reflections[i]
on the right.This acts by switching the entries \(i\) and \(i+1\).
Warning
This ignores the multiplication convention in order to be consistent with other Coxeter operations in permutations (e.g., computing
reduced_word()
).EXAMPLES:
sage: W = Permutations(3) sage: w = W([2,3,1]) sage: w.apply_simple_reflection_right(1) [3, 2, 1] sage: w.apply_simple_reflection_right(2) [2, 1, 3]
 has_left_descent(i, mult=None)¶
Check if
i
is a left descent ofself
.A left descent of a permutation \(\pi \in S_n\) means an index \(i \in \{ 1, 2, \ldots, n1 \}\) such that \(s_i \circ \pi\) has smaller length than \(\pi\). Thus, a left descent of \(\pi\) is an index \(i \in \{ 1, 2, \ldots, n1 \}\) satisfying \(\pi^{1}(i) > \pi^{1}(i+1)\).
Warning
The methods
descents()
andidescents()
behave differently than their Weyl group counterparts. In particular, the indexing is 0based. This could lead to errors. Instead, construct the descent set as in the example.Warning
This ignores the multiplication convention in order to be consistent with other Coxeter operations in permutations (e.g., computing
reduced_word()
).EXAMPLES:
sage: P = Permutations(4) sage: x = P([3, 2, 4, 1]) sage: (~x).descents() [1, 2] sage: [i for i in P.index_set() if x.has_left_descent(i)] [1, 2]
 has_right_descent(i, mult=None)¶
Check if
i
is a right descent ofself
.A right descent of a permutation \(\pi \in S_n\) means an index \(i \in \{ 1, 2, \ldots, n1 \}\) such that \(\pi \circ s_i\) has smaller length than \(\pi\). Thus, a right descent of \(\pi\) is an index \(i \in \{ 1, 2, \ldots, n1 \}\) satisfying \(\pi(i) > \pi(i+1)\).
Warning
The methods
descents()
andidescents()
behave differently than their Weyl group counterparts. In particular, the indexing is 0based. This could lead to errors. Instead, construct the descent set as in the example.Warning
This ignores the multiplication convention in order to be consistent with other Coxeter operations in permutations (e.g., computing
reduced_word()
).EXAMPLES:
sage: P = Permutations(4) sage: x = P([3, 2, 4, 1]) sage: x.descents() [1, 3] sage: [i for i in P.index_set() if x.has_right_descent(i)] [1, 3]
 inverse()¶
Return the inverse of
self
.EXAMPLES:
sage: P = Permutations(4) sage: w0 = P([4,3,2,1]) sage: w0.inverse() == w0 True sage: w0.inverse().parent() is P True sage: P([3,2,4,1]).inverse() [4, 2, 1, 3]
 algebra(base_ring, category=None)¶
Return the symmetric group algebra associated to
self
.INPUT:
base_ring
– a ringcategory
– a category (default: the category ofself
)
EXAMPLES:
sage: P = Permutations(4) sage: A = P.algebra(QQ); A Symmetric group algebra of order 4 over Rational Field sage: A.category() Join of Category of coxeter group algebras over Rational Field and Category of finite group algebras over Rational Field and Category of finite dimensional cellular algebras with basis over Rational Field sage: A = P.algebra(QQ, category=Monoids()) sage: A.category() Category of finite dimensional cellular monoid algebras over Rational Field
 as_permutation_group()¶
Return
self
as a permutation group.EXAMPLES:
sage: P = Permutations(4) sage: PG = P.as_permutation_group() sage: PG Symmetric group of order 4! as a permutation group sage: G = SymmetricGroup(4) sage: PG is G True
 cardinality()¶
Return the number of permutations of size \(n\), which is \(n!\).
EXAMPLES:
sage: Permutations(0).cardinality() 1 sage: Permutations(3).cardinality() 6 sage: Permutations(4).cardinality() 24
 cartan_type()¶
Return the Cartan type of
self
.The symmetric group \(S_n\) is a Coxeter group of type \(A_{n1}\).
EXAMPLES:
sage: A = SymmetricGroup([2,3,7]); A.cartan_type() ['A', 2] sage: A = SymmetricGroup([]); A.cartan_type() ['A', 0]
 codegrees()¶
Return the codegrees of
self
.EXAMPLES:
sage: Permutations(3).codegrees() (0, 1) sage: Permutations(7).codegrees() (0, 1, 2, 3, 4, 5)
 conjugacy_class(g)¶
Return the conjugacy class of
g
inself
.INPUT:
g
– a partition or an element ofself
EXAMPLES:
sage: G = Permutations(5) sage: g = G([2,3,4,1,5]) sage: G.conjugacy_class(g) Conjugacy class of cycle type [4, 1] in Standard permutations of 5 sage: G.conjugacy_class(Partition([2, 1, 1, 1])) Conjugacy class of cycle type [2, 1, 1, 1] in Standard permutations of 5
 conjugacy_classes()¶
Return a list of the conjugacy classes of
self
.EXAMPLES:
sage: G = Permutations(4) sage: G.conjugacy_classes() [Conjugacy class of cycle type [1, 1, 1, 1] in Standard permutations of 4, Conjugacy class of cycle type [2, 1, 1] in Standard permutations of 4, Conjugacy class of cycle type [2, 2] in Standard permutations of 4, Conjugacy class of cycle type [3, 1] in Standard permutations of 4, Conjugacy class of cycle type [4] in Standard permutations of 4]
 conjugacy_classes_iterator()¶
Iterate over the conjugacy classes of
self
.EXAMPLES:
sage: G = Permutations(4) sage: list(G.conjugacy_classes_iterator()) == G.conjugacy_classes() True
 conjugacy_classes_representatives()¶
Return a complete list of representatives of conjugacy classes in
self
.Let \(S_n\) be the symmetric group on \(n\) letters. The conjugacy classes are indexed by partitions \(\lambda\) of \(n\). The ordering of the conjugacy classes is reverse lexicographic order of the partitions.
EXAMPLES:
sage: G = Permutations(5) sage: G.conjugacy_classes_representatives() [[1, 2, 3, 4, 5], [2, 1, 3, 4, 5], [2, 1, 4, 3, 5], [2, 3, 1, 4, 5], [2, 3, 1, 5, 4], [2, 3, 4, 1, 5], [2, 3, 4, 5, 1]]
 degree()¶
Return the degree of
self
.This is the cardinality \(n\) of the set
self
acts on.EXAMPLES:
sage: Permutations(0).degree() 0 sage: Permutations(1).degree() 1 sage: Permutations(5).degree() 5
 degrees()¶
Return the degrees of
self
.These are the degrees of the fundamental invariants of the ring of polynomial invariants.
EXAMPLES:
sage: Permutations(3).degrees() (2, 3) sage: Permutations(7).degrees() (2, 3, 4, 5, 6, 7)
 element_in_conjugacy_classes(nu)¶
Return a permutation with cycle type
nu
.If the size of
nu
is smaller than the size of permutations inself
, then some fixed points are added.EXAMPLES:
sage: PP = Permutations(5) sage: PP.element_in_conjugacy_classes([2,2]) [2, 1, 4, 3, 5]
 identity()¶
Return the identity permutation of size \(n\).
EXAMPLES:
sage: Permutations(4).identity() [1, 2, 3, 4] sage: Permutations(0).identity() []
 index_set()¶
Return the index set for the descents of the symmetric group
self
.This is \(\{ 1, 2, \ldots, n1 \}\), where
self
is \(S_n\).EXAMPLES:
sage: P = Permutations(8) sage: P.index_set() (1, 2, 3, 4, 5, 6, 7)
 one()¶
Return the identity permutation of size \(n\).
EXAMPLES:
sage: Permutations(4).identity() [1, 2, 3, 4] sage: Permutations(0).identity() []
 random_element()¶
EXAMPLES:
sage: s = Permutations(4).random_element(); s # random [1, 2, 4, 3] sage: s in Permutations(4) True
 rank(p=None)¶
Return the rank of
self
orp
depending on input.If a permutation
p
is given, return the rank ofp
inself
. Otherwise return the dimension of the underlying vector space spanned by the (simple) roots.EXAMPLES:
sage: P = Permutations(5) sage: P.rank() 4 sage: SP3 = Permutations(3) sage: list(map(SP3.rank, SP3)) [0, 1, 2, 3, 4, 5] sage: SP0 = Permutations(0) sage: list(map(SP0.rank, SP0)) [0]
 simple_reflection(i)¶
For \(i\) in the index set of
self
(that is, for \(i\) in \(\{ 1, 2, \ldots, n1 \}\), whereself
is \(S_n\)), this returns the elementary transposition \(s_i = (i,i+1)\).EXAMPLES:
sage: P = Permutations(4) sage: P.simple_reflection(2) [1, 3, 2, 4] sage: P.simple_reflections() Finite family {1: [2, 1, 3, 4], 2: [1, 3, 2, 4], 3: [1, 2, 4, 3]}
 unrank(r)¶
EXAMPLES:
sage: SP3 = Permutations(3) sage: l = list(map(SP3.unrank, range(6))) sage: l == SP3.list() True sage: SP0 = Permutations(0) sage: l = list(map(SP0.unrank, range(1))) sage: l == SP0.list() True
 class sage.combinat.permutation.StandardPermutations_n_abstract(n, category=None)¶
Bases:
sage.combinat.permutation.Permutations
Abstract base class for subsets of permutations of the set \(\{1, 2, \ldots, n\}\).
Warning
Anything inheriting from this class should override the
__contains__
method.
 class sage.combinat.permutation.StandardPermutations_recoils(recoils)¶
Bases:
sage.combinat.permutation.Permutations
Permutations of \(\{1, \ldots, n\}\) with a fixed recoils composition.
 class sage.combinat.permutation.StandardPermutations_recoilsfatter(recoils)¶
 class sage.combinat.permutation.StandardPermutations_recoilsfiner(recoils)¶
 sage.combinat.permutation.bistochastic_as_sum_of_permutations(M, check=True)¶
Return the positive sum of permutations corresponding to the bistochastic matrix
M
.A stochastic matrix is a matrix with nonnegative real entries such that the sum of the elements of any row is equal to \(1\). A bistochastic matrix is a stochastic matrix whose transpose matrix is also stochastic ( there are conditions both on the rows and on the columns ).
According to the Birkhoffvon Neumann Theorem, any bistochastic matrix can be written as a convex combination of permutation matrices, which also means that the polytope of bistochastic matrices is integer.
As a nonbistochastic matrix can obviously not be written as a convex combination of permutations, this theorem is an equivalence.
This function, given a bistochastic matrix, returns the corresponding decomposition.
INPUT:
M
– A bistochastic matrixcheck
(boolean) – set toTrue
(default) to check that the matrix is indeed bistochastic
OUTPUT:
An element of
CombinatorialFreeModule
, which is a free \(F\)module ( where \(F\) is the ground ring of the given matrix ) whose basis is indexed by the permutations.
Note
In this function, we just assume 1 to be any constant : for us a matrix \(M\) is bistochastic if there exists \(c>0\) such that \(M/c\) is bistochastic.
You can obtain a sequence of pairs
(permutation,coeff)
, wherepermutation
is a SagePermutation
instance, andcoeff
its corresponding coefficient from the result of this function by applying thelist
function.If you are interested in the matrix corresponding to a
Permutation
you will be glad to learn about thePermutation.to_matrix()
method.The base ring of the matrix can be anything that can be coerced to
RR
.
See also
as_sum_of_permutations()
to use this method through theMatrix
class.
EXAMPLES:
We create a bistochastic matrix from a convex sum of permutations, then try to deduce the decomposition from the matrix:
sage: from sage.combinat.permutation import bistochastic_as_sum_of_permutations sage: L = [] sage: L.append((9,Permutation([4, 1, 3, 5, 2]))) sage: L.append((6,Permutation([5, 3, 4, 1, 2]))) sage: L.append((3,Permutation([3, 1, 4, 2, 5]))) sage: L.append((2,Permutation([1, 4, 2, 3, 5]))) sage: M = sum([c * p.to_matrix() for (c,p) in L]) sage: decomp = bistochastic_as_sum_of_permutations(M) sage: print(decomp) 2*B[[1, 4, 2, 3, 5]] + 3*B[[3, 1, 4, 2, 5]] + 9*B[[4, 1, 3, 5, 2]] + 6*B[[5, 3, 4, 1, 2]]
An exception is raised when the matrix is not positive and bistochastic:
sage: M = Matrix([[2,3],[2,2]]) sage: decomp = bistochastic_as_sum_of_permutations(M) Traceback (most recent call last): ... ValueError: The matrix is not bistochastic sage: bistochastic_as_sum_of_permutations(Matrix(GF(7), 2, [2,1,1,2])) Traceback (most recent call last): ... ValueError: The base ring of the matrix must have a coercion map to RR sage: bistochastic_as_sum_of_permutations(Matrix(ZZ, 2, [2,1,1,2])) Traceback (most recent call last): ... ValueError: The matrix should have nonnegative entries
 sage.combinat.permutation.bounded_affine_permutation(A)¶
Return the bounded affine permutation of a matrix.
The bounded affine permutation of a matrix \(A\) with entries in \(R\) is a partial permutation of length \(n\), where \(n\) is the number of columns of \(A\). The entry in position \(i\) is the smallest value \(j\) such that column \(i\) is in the span of columns \(i+1, \ldots, j\), over \(R\), where column indices are taken modulo \(n\). If column \(i\) is the zero vector, then the permutation has a fixed point at \(i\).
INPUT:
A
– matrix with entries in a ring \(R\)
EXAMPLES:
sage: from sage.combinat.permutation import bounded_affine_permutation sage: A = Matrix(ZZ, [[1,0,0,0], [0,1,0,0]]) sage: bounded_affine_permutation(A) [5, 6, 3, 4] sage: A = Matrix(ZZ, [[0,1,0,1,0], [0,0,1,1,0]]) sage: bounded_affine_permutation(A) [1, 4, 7, 8, 5]
REFERENCES:
 sage.combinat.permutation.bruhat_lequal(p1, p2)¶
Return
True
ifp1
is less thanp2
in the Bruhat order.Algorithm from mupadcombinat.
EXAMPLES:
sage: import sage.combinat.permutation as permutation sage: permutation.bruhat_lequal([2,4,3,1],[3,4,2,1]) True
 sage.combinat.permutation.descents_composition_first(dc)¶
Compute the smallest element of a descent class having a descent composition
dc
.EXAMPLES:
sage: import sage.combinat.permutation as permutation sage: permutation.descents_composition_first([1,1,3,4,3]) [3, 2, 1, 4, 6, 5, 7, 8, 10, 9, 11, 12]
 sage.combinat.permutation.descents_composition_last(dc)¶
Return the largest element of a descent class having a descent composition
dc
.EXAMPLES:
sage: import sage.combinat.permutation as permutation sage: permutation.descents_composition_last([1,1,3,4,3]) [12, 11, 8, 9, 10, 4, 5, 6, 7, 1, 2, 3]
 sage.combinat.permutation.descents_composition_list(dc)¶
Return a list of all the permutations that have the descent composition
dc
.EXAMPLES:
sage: import sage.combinat.permutation as permutation sage: permutation.descents_composition_list([1,2,2]) [[5, 2, 4, 1, 3], [5, 3, 4, 1, 2], [4, 3, 5, 1, 2], [4, 2, 5, 1, 3], [3, 2, 5, 1, 4], [2, 1, 5, 3, 4], [3, 1, 5, 2, 4], [4, 1, 5, 2, 3], [5, 1, 4, 2, 3], [5, 1, 3, 2, 4], [4, 1, 3, 2, 5], [3, 1, 4, 2, 5], [2, 1, 4, 3, 5], [3, 2, 4, 1, 5], [4, 2, 3, 1, 5], [5, 2, 3, 1, 4]]
 sage.combinat.permutation.from_cycles(n, cycles, parent=None)¶
Return the permutation in the \(n\)th symmetric group whose decomposition into disjoint cycles is
cycles
.This function checks that its input is correct (i.e. that the cycles are disjoint and their elements integers among \(1...n\)). It raises an exception otherwise.
Warning
It assumes that the elements are of
int
type.EXAMPLES:
sage: import sage.combinat.permutation as permutation sage: permutation.from_cycles(4, [[1,2]]) [2, 1, 3, 4] sage: permutation.from_cycles(4, [[1,2,4]]) [2, 4, 3, 1] sage: permutation.from_cycles(10, [[3,1],[4,5],[6,8,9]]) [3, 2, 1, 5, 4, 8, 7, 9, 6, 10] sage: permutation.from_cycles(10, ((2, 5), (6, 1, 3))) [3, 5, 6, 4, 2, 1, 7, 8, 9, 10] sage: permutation.from_cycles(4, []) [1, 2, 3, 4] sage: permutation.from_cycles(4, [[]]) [1, 2, 3, 4] sage: permutation.from_cycles(0, []) []
Bad input (see trac ticket #13742):
sage: Permutation("(12,2)(3,4)") Traceback (most recent call last): ... ValueError: All elements should be strictly positive integers, and I just found a nonpositive one. sage: Permutation("(1,2)(2,4)") Traceback (most recent call last): ... ValueError: an element appears twice in the input sage: permutation.from_cycles(4, [[1,18]]) Traceback (most recent call last): ... ValueError: You claimed that this was a permutation on 1...4 but it contains 18
 sage.combinat.permutation.from_inversion_vector(iv, parent=None)¶
Return the permutation corresponding to inversion vector
iv
.See \(~sage.combinat.permutation.Permutation.to_inversion_vector\) for a definition of the inversion vector of a permutation.
EXAMPLES:
sage: import sage.combinat.permutation as permutation sage: permutation.from_inversion_vector([3,1,0,0,0]) [3, 2, 4, 1, 5] sage: permutation.from_inversion_vector([2,3,6,4,0,2,2,1,0]) [5, 9, 1, 8, 2, 6, 4, 7, 3] sage: permutation.from_inversion_vector([0]) [1] sage: permutation.from_inversion_vector([]) []
 sage.combinat.permutation.from_lehmer_code(lehmer, parent=None)¶
Return the permutation with Lehmer code
lehmer
.EXAMPLES:
sage: import sage.combinat.permutation as permutation sage: lc = Permutation([2,1,5,4,3]).to_lehmer_code(); lc [1, 0, 2, 1, 0] sage: permutation.from_lehmer_code(lc) [2, 1, 5, 4, 3]
 sage.combinat.permutation.from_major_code(mc, final_descent=False)¶
Return the permutation with major code
mc
.The major code of a permutation is defined in
to_major_code()
.Warning
This function creates illegal permutations (i.e.
Permutation([9])
, and this is dangerous as thePermutation()
class is only designed to handle permutations on \(1...n\). This will have to be changed when Sage permutations will be able to handle anything, but right now this should be fixed. Be careful with the results.Warning
If
mc
is not a major index of a permutation, then the return value of this method can be anything. Garbage in, garbage out!REFERENCES:
Skandera, M. An Eulerian Partner for Inversions. Sem. Lothar. Combin. 46 (2001) B46d.
EXAMPLES:
sage: import sage.combinat.permutation as permutation sage: permutation.from_major_code([5, 0, 1, 0, 1, 2, 0, 1, 0]) [9, 3, 5, 7, 2, 1, 4, 6, 8] sage: permutation.from_major_code([8, 3, 3, 1, 4, 0, 1, 0, 0]) [2, 8, 4, 3, 6, 7, 9, 5, 1] sage: Permutation([2,1,6,4,7,3,5]).to_major_code() [3, 2, 0, 2, 2, 0, 0] sage: permutation.from_major_code([3, 2, 0, 2, 2, 0, 0]) [2, 1, 6, 4, 7, 3, 5]
 sage.combinat.permutation.from_permutation_group_element(pge, parent=None)¶
Return a
Permutation
given aPermutationGroupElement
pge
.EXAMPLES:
sage: import sage.combinat.permutation as permutation sage: pge = PermutationGroupElement([(1,2),(3,4)]) sage: permutation.from_permutation_group_element(pge) [2, 1, 4, 3]
 sage.combinat.permutation.from_rank(n, rank)¶
Return the permutation of the set \(\{1,...,n\}\) with lexicographic rank
rank
. This is the permutation whose Lehmer code is the factoradic representation ofrank
. In particular, the permutation with rank \(0\) is the identity permutation.The permutation is computed without iterating through all of the permutations with lower rank. This makes it efficient for large permutations.
Note
The variable
rank
is not checked for being in the interval from \(0\) to \(n!  1\). When outside this interval, it acts as its residue modulo \(n!\).EXAMPLES:
sage: import sage.combinat.permutation as permutation sage: Permutation([3, 6, 5, 4, 2, 1]).rank() 359 sage: [permutation.from_rank(3, i) for i in range(6)] [[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]] sage: Permutations(6)[10] [1, 2, 4, 6, 3, 5] sage: permutation.from_rank(6,10) [1, 2, 4, 6, 3, 5]
 sage.combinat.permutation.from_reduced_word(rw, parent=None)¶
Return the permutation corresponding to the reduced word
rw
.See
reduced_words()
for a definition of reduced words and the convention on the order of multiplication used.EXAMPLES:
sage: import sage.combinat.permutation as permutation sage: permutation.from_reduced_word([3,2,3,1,2,3,1]) [3, 4, 2, 1] sage: permutation.from_reduced_word([]) []
 sage.combinat.permutation.permutohedron_lequal(p1, p2, side='right')¶
Return
True
ifp1
is less than 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.EXAMPLES:
sage: import sage.combinat.permutation as permutation sage: permutation.permutohedron_lequal(Permutation([3,2,1,4]),Permutation([4,2,1,3])) False sage: permutation.permutohedron_lequal(Permutation([3,2,1,4]),Permutation([4,2,1,3]), side='left') True
 sage.combinat.permutation.to_standard(p, key=None)¶
Return a standard permutation corresponding to the iterable
p
.INPUT:
p
– an iterablekey
– (optional) a comparison key for the elementx
ofp
EXAMPLES:
sage: import sage.combinat.permutation as permutation sage: permutation.to_standard([4,2,7]) [2, 1, 3] sage: permutation.to_standard([1,2,3]) [1, 2, 3] sage: permutation.to_standard([]) [] sage: permutation.to_standard([1,2,3], key=lambda x: x) [3, 2, 1] sage: permutation.to_standard([5,8,2,5], key=lambda x: x) [2, 1, 4, 3]