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 github issue #13742). This is dangerous. In particular, the Permutation._left_to_right_multiply_on_right() method (which can be called through multiplication) disables the input checks (see Permutation()). This should not happen. Do not trust the results.

What does this file define ?#

The main part of this file consists in the definition of permutation objects, i.e. the Permutation() method and the Permutation class. Global options for elements of the permutation class can be set through the Permutations.options() object.

Below are listed all methods and classes defined in this file.

Methods of Permutations objects

`left_action_product()`	Returns the product of `self` with another permutation, in which the other permutation is applied first.
`right_action_product()`	Returns the product of `self` with another permutation, in which `self` is applied first.
`size()`	Returns the size of the permutation `self`.
`cycle_string()`	Returns the disjoint-cycles representation of `self` as string.
`next()`	Returns the permutation that follows `self` in lexicographic order (in the same symmetric group as `self`).
`prev()`	Returns the permutation that comes directly before `self` in lexicographic order (in the same symmetric group as `self`).
`to_tableau_by_shape()`	Returns a tableau of shape `shape` with the entries in `self`.
`to_cycles()`	Returns the permutation `self` as a list of disjoint cycles.
`forget_cycles()`	Return `self` under the forget cycle map.
`to_permutation_group_element()`	Returns a `PermutationGroupElement` equal to `self`.
`signature()`	Returns the signature of the permutation `sef`.
`is_even()`	Returns `True` if the permutation `self` is even, and `False` otherwise.
`to_matrix()`	Returns a matrix representing the permutation `self`.
`rank()`	Returns the rank of `self` in lexicographic ordering (on the symmetric group containing `self`).
`to_inversion_vector()`	Returns the inversion vector of a permutation `self`.
`inversions()`	Returns a list of the inversions of permutation `self`.
`stack_sort()`	Returns the permutation obtained by sorting `self` through one stack.
`to_digraph()`	Return a digraph representation of `self`.
`show()`	Displays the permutation as a drawing.
`number_of_inversions()`	Returns the number of inversions in the permutation `self`.
`noninversions()`	Returns the `k`-noninversions in the permutation `self`.
`number_of_noninversions()`	Returns the number of `k`-noninversions in the permutation `self`.
`length()`	Returns the Coxeter length of a permutation `self`.
`inverse()`	Returns the inverse of a permutation `self`.
`ishift()`	Returns the `i`-shift of `self`.
`iswitch()`	Returns the `i`-switch of `self`.
`runs()`	Returns a list of the runs in the permutation `self`.
`longest_increasing_subsequence_length()`	Returns the length of the longest increasing subsequences of `self`.
`longest_increasing_subsequences()`	Returns the list of the longest increasing subsequences of `self`.
`longest_increasing_subsequences_number()`	Returns the number of longest increasing subsequences
`cycle_type()`	Returns the cycle type of `self` as a partition of `len(self)`.
`foata_bijection()`	Returns the image of the permutation `self` under the Foata bijection \(\phi\).
`foata_bijection_inverse()`	Returns the image of the permutation `self` under the inverse of the Foata bijection \(\phi\).
`fundamental_transformation()`	Returns the image of the permutation `self` under the Renyi-Foata-Schuetzenberger fundamental transformation.
`fundamental_transformation_inverse()`	Returns the image of the permutation `self` under the inverse of the Renyi-Foata-Schuetzenberger fundamental transformation.
`destandardize()`	Return destandardization of `self` with respect to `weight` and `ordered_alphabet`.
`to_lehmer_code()`	Returns the Lehmer code of the permutation `self`.
`to_lehmer_cocode()`	Returns the Lehmer cocode of `self`.
`reduced_word()`	Returns the reduced word of the permutation `self`.
`reduced_words()`	Returns a list of the reduced words of the permutation `self`.
`reduced_words_iterator()`	An iterator for the reduced words of the permutation `self`.
`reduced_word_lexmin()`	Returns a lexicographically minimal reduced word of a permutation `self`.
`fixed_points()`	Returns a list of the fixed points of the permutation `self`.
`is_derangement()`	Returns `True` if the permutation `self` is a derangement, and `False` otherwise.
`is_simple()`	Returns `True` if the permutation `self` is simple, and `False` otherwise.
`number_of_fixed_points()`	Returns the number of fixed points of the permutation `self`.
`recoils()`	Returns the list of the positions of the recoils of the permutation `self`.
`number_of_recoils()`	Returns the number of recoils of the permutation `self`.
`recoils_composition()`	Returns the composition corresponding to the recoils of `self`.
`descents()`	Returns the list of the descents of the permutation `self`.
`idescents()`	Returns a list of the idescents of `self`.
`idescents_signature()`	Returns the list obtained by mapping each position in `self` to \(-1\) if it is an idescent and \(1\) if it is not an idescent.
`number_of_descents()`	Returns the number of descents of the permutation `self`.
`number_of_idescents()`	Returns the number of idescents of the permutation `self`.
`descents_composition()`	Returns the composition corresponding to the descents of `self`.
`descent_polynomial()`	Returns the descent polynomial of the permutation `self`.
`major_index()`	Returns the major index of the permutation `self`.
`imajor_index()`	Returns the inverse major index of the permutation `self`.
`to_major_code()`	Returns the major code of the permutation `self`.
`peaks()`	Returns a list of the peaks of the permutation `self`.
`number_of_peaks()`	Returns the number of peaks of the permutation `self`.
`saliances()`	Returns a list of the saliances of the permutation `self`.
`number_of_saliances()`	Returns the number of saliances of the permutation `self`.
`bruhat_lequal()`	Returns `True` if self is less or equal to `p2` in the Bruhat order.
`weak_excedences()`	Returns all the numbers `self[i]` such that `self[i] >= i+1`.
`bruhat_inversions()`	Returns the list of inversions of `self` such that the application of this inversion to `self` decrements its number of inversions.
`bruhat_inversions_iterator()`	Returns an iterator over Bruhat inversions of `self`.
`bruhat_succ()`	Returns a list of the permutations covering `self` in the Bruhat order.
`bruhat_succ_iterator()`	An iterator for the permutations covering `self` in the Bruhat order.
`bruhat_pred()`	Returns a list of the permutations covered by `self` in the Bruhat order.
`bruhat_pred_iterator()`	An iterator for the permutations covered by `self` in the Bruhat order.
`bruhat_smaller()`	Returns the combinatorial class of permutations smaller than or equal to `self` in the Bruhat order.
`bruhat_greater()`	Returns the combinatorial class of permutations greater than or equal to `self` in the Bruhat order.
`permutohedron_lequal()`	Returns `True` if `self` is less or equal to `p2` in the permutohedron order.
`permutohedron_succ()`	Returns a list of the permutations covering `self` in the permutohedron order.
`permutohedron_pred()`	Returns a list of the permutations covered by `self` in the permutohedron order.
`permutohedron_smaller()`	Returns a list of permutations smaller than or equal to `self` in the permutohedron order.
`permutohedron_greater()`	Returns a list of permutations greater than or equal to `self` in the permutohedron order.
`right_permutohedron_interval_iterator()`	Returns an iterator over permutations in an interval of the permutohedron order.
`right_permutohedron_interval()`	Returns a list of permutations in an interval of the permutohedron order.
`has_pattern()`	Tests whether the permutation `self` matches the pattern.
`avoids()`	Tests whether the permutation `self` avoids the pattern.
`pattern_positions()`	Returns the list of positions where the pattern `patt` appears in `self`.
`reverse()`	Returns the permutation obtained by reversing the 1-line notation of `self`.
`complement()`	Returns the complement of the permutation which is obtained by replacing each value \(x\) in the 1-line notation of `self` with \(n - x + 1\).
`permutation_poset()`	Returns the permutation poset of `self`.
`dict()`	Returns a dictionary corresponding to the permutation `self`.
`action()`	Returns the action of the permutation `self` on a list.
`robinson_schensted()`	Returns the pair of standard tableaux obtained by running the Robinson-Schensted Algorithm on `self`.
`left_tableau()`	Returns the left standard tableau after performing the RSK algorithm.
`right_tableau()`	Returns the right standard tableau after performing the RSK algorithm.
`increasing_tree()`	Returns the increasing tree of `self`.
`increasing_tree_shape()`	Returns the shape of the increasing tree of `self`.
`binary_search_tree()`	Returns the binary search tree of `self`.
`sylvester_class()`	Iterates over the equivalence class of `self` under sylvester congruence
`RS_partition()`	Returns the shape of the tableaux obtained by the RSK algorithm.
`remove_extra_fixed_points()`	Returns the permutation obtained by removing any fixed points at the end of `self`.
`retract_plain()`	Returns the plain retract of `self` to a smaller symmetric group \(S_m\).
`retract_direct_product()`	Returns the direct-product retract of `self` to a smaller symmetric group \(S_m\).
`retract_okounkov_vershik()`	Returns the Okounkov-Vershik retract of `self` to a smaller symmetric group \(S_m\).
`hyperoctahedral_double_coset_type()`	Returns the coset-type of `self` as a partition.
`binary_search_tree_shape()`	Returns the shape of the binary search tree of `self` (a non labelled binary tree).
`shifted_concatenation()`	Returns the right (or left) shifted concatenation of `self` with a permutation `other`.
`shifted_shuffle()`	Returns the shifted shuffle of `self` with a permutation `other`.

Other classes defined in this file

`Permutations`
`Permutations_nk`
`Permutations_mset`
`Permutations_set`
`Permutations_msetk`
`Permutations_setk`
`Arrangements`
`Arrangements_msetk`
`Arrangements_setk`
`StandardPermutations_all`
`StandardPermutations_n_abstract`
`StandardPermutations_n`
`StandardPermutations_descents`
`StandardPermutations_recoilsfiner`
`StandardPermutations_recoilsfatter`
`StandardPermutations_recoils`
`StandardPermutations_bruhat_smaller`
`StandardPermutations_bruhat_greater`
`CyclicPermutations`
`CyclicPermutationsOfPartition`
`StandardPermutations_avoiding_12`
`StandardPermutations_avoiding_21`
`StandardPermutations_avoiding_132`
`StandardPermutations_avoiding_123`
`StandardPermutations_avoiding_321`
`StandardPermutations_avoiding_231`
`StandardPermutations_avoiding_312`
`StandardPermutations_avoiding_213`
`StandardPermutations_avoiding_generic`
`PatternAvoider`

Functions defined in this file

`from_major_code()`	Returns the permutation corresponding to major code `mc`.
`from_permutation_group_element()`	Returns a Permutation give a `PermutationGroupElement` `pge`.
`from_rank()`	Returns the permutation with the specified lexicographic rank.
`from_inversion_vector()`	Returns the permutation corresponding to inversion vector `iv`.
`from_cycles()`	Returns the permutation with given disjoint-cycle representation `cycles`.
`from_lehmer_code()`	Returns the permutation with Lehmer code `lehmer`.
`from_reduced_word()`	Returns the permutation corresponding to the reduced word `rw`.
`bistochastic_as_sum_of_permutations()`	Returns a given bistochastic matrix as a nonnegative linear combination of permutations.
`bounded_affine_permutation()`	Returns a partial permutation representing the bounded affine permutation of a matrix.
`descents_composition_list()`	Returns a list of all the permutations in a given descent class (i. e., having a given descents composition).
`descents_composition_first()`	Returns the smallest element of a descent class.
`descents_composition_last()`	Returns the largest element of a descent class.
`bruhat_lequal()`	Returns `True` if `p1` is less or equal to `p2` in the Bruhat order.
`permutohedron_lequal()`	Returns `True` if `p1` is less or equal to `p2` in the permutohedron order.
`to_standard()`	Returns a standard permutation corresponding to the permutation `self`.

AUTHORS:

Mike Hansen
Dan Drake (2008-04-07): allow Permutation() to take lists of tuples
Sébastien Labbé (2009-03-17): added robinson_schensted_inverse
Travis Scrimshaw:
- (2012-08-16): to_standard() no longer modifies input
- (2013-01-19): Removed RSK implementation and moved to rsk.
- (2013-07-13): Removed CombinatorialClass and moved permutations to the category framework.
Darij Grinberg (2013-09-07): added methods; ameliorated github issue #14885 by exposing and documenting methods for global-independent multiplication.
Travis Scrimshaw (2014-02-05): Made StandardPermutations_n a finite Weyl group to make it more uniform with SymmetricGroup. Added ability to compute the conjugacy classes.
Trevor K. Karn (2022-08-05): Add Permutation.n_reduced_words()
Amrutha P, Shriya M, Divya Aggarwal (2022-08-16): Added Multimajor Index.

Classes and methods#

class sage.combinat.permutation.Arrangements#

Bases: Permutations

An arrangement of a multiset mset is an ordered selection without repetitions. It is represented by a list that contains only elements from mset, but maybe in a different order.

Arrangements returns the combinatorial class of arrangements of the multiset mset that contain k elements.

EXAMPLES:

sage: mset = [1,1,2,3,4,4,5]
sage: Arrangements(mset, 2).list()                                              # needs sage.libs.gap
[[1, 1],
 [1, 2],
 [1, 3],
 [1, 4],
 [1, 5],
 [2, 1],
 [2, 3],
 [2, 4],
 [2, 5],
 [3, 1],
 [3, 2],
 [3, 4],
 [3, 5],
 [4, 1],
 [4, 2],
 [4, 3],
 [4, 4],
 [4, 5],
 [5, 1],
 [5, 2],
 [5, 3],
 [5, 4]]
 sage: Arrangements(mset, 2).cardinality()                                      # needs sage.libs.gap
 22
 sage: Arrangements( ["c","a","t"], 2 ).list()
 [['c', 'a'], ['c', 't'], ['a', 'c'], ['a', 't'], ['t', 'c'], ['t', 'a']]
 sage: Arrangements( ["c","a","t"], 3 ).list()
 [['c', 'a', 't'],
  ['c', 't', 'a'],
  ['a', 'c', 't'],
  ['a', 't', 'c'],
  ['t', 'c', 'a'],
  ['t', 'a', 'c']]

cardinality()#

Return the cardinality of self.

EXAMPLES:

sage: A = Arrangements([1,1,2,3,4,4,5], 2)
sage: A.cardinality()                                                       # needs sage.libs.gap
22

class sage.combinat.permutation.Arrangements_msetk(mset, k)#

Bases: Arrangements, Permutations_msetk

Arrangements of length \(k\) of a multiset \(M\).

class sage.combinat.permutation.Arrangements_setk(s, k)#

Bases: Arrangements, Permutations_setk

Arrangements of length \(k\) of a set \(S\).

class sage.combinat.permutation.CyclicPermutations(mset)#

Bases: Permutations_mset

Return the class of all cyclic permutations of mset in cycle notation. These are the same as necklaces.

INPUT:

mset – A multiset

EXAMPLES:

sage: CyclicPermutations(range(4)).list()                                       # needs sage.combinat
[[0, 1, 2, 3],
 [0, 1, 3, 2],
 [0, 2, 1, 3],
 [0, 2, 3, 1],
 [0, 3, 1, 2],
 [0, 3, 2, 1]]
sage: CyclicPermutations([1,1,1]).list()                                        # needs sage.combinat
[[1, 1, 1]]

iterator(distinct=False)#

EXAMPLES:

sage: CyclicPermutations(range(4)).list()  # indirect doctest               # needs sage.combinat
[[0, 1, 2, 3],
 [0, 1, 3, 2],
 [0, 2, 1, 3],
 [0, 2, 3, 1],
 [0, 3, 1, 2],
 [0, 3, 2, 1]]
 sage: CyclicPermutations([1,1,1]).list()                                   # needs sage.combinat
 [[1, 1, 1]]
 sage: CyclicPermutations([1,1,1]).list(distinct=True)                      # needs sage.combinat
 [[1, 1, 1], [1, 1, 1]]

list(distinct=False)#

EXAMPLES:

sage: CyclicPermutations(range(4)).list()                                   # needs sage.combinat
[[0, 1, 2, 3],
 [0, 1, 3, 2],
 [0, 2, 1, 3],
 [0, 2, 3, 1],
 [0, 3, 1, 2],
 [0, 3, 2, 1]]

class sage.combinat.permutation.CyclicPermutationsOfPartition(partition)#

Bases: Permutations

Combinations of cyclic permutations of each cell of a given partition.

This is the same as a Cartesian product of necklaces.

EXAMPLES:

sage: CyclicPermutationsOfPartition([[1,2,3,4],[5,6,7]]).list()                 # needs sage.combinat
[[[1, 2, 3, 4], [5, 6, 7]],
 [[1, 2, 4, 3], [5, 6, 7]],
 [[1, 3, 2, 4], [5, 6, 7]],
 [[1, 3, 4, 2], [5, 6, 7]],
 [[1, 4, 2, 3], [5, 6, 7]],
 [[1, 4, 3, 2], [5, 6, 7]],
 [[1, 2, 3, 4], [5, 7, 6]],
 [[1, 2, 4, 3], [5, 7, 6]],
 [[1, 3, 2, 4], [5, 7, 6]],
 [[1, 3, 4, 2], [5, 7, 6]],
 [[1, 4, 2, 3], [5, 7, 6]],
 [[1, 4, 3, 2], [5, 7, 6]]]

sage: CyclicPermutationsOfPartition([[1,2,3,4],[4,4,4]]).list()                 # needs sage.combinat
[[[1, 2, 3, 4], [4, 4, 4]],
 [[1, 2, 4, 3], [4, 4, 4]],
 [[1, 3, 2, 4], [4, 4, 4]],
 [[1, 3, 4, 2], [4, 4, 4]],
 [[1, 4, 2, 3], [4, 4, 4]],
 [[1, 4, 3, 2], [4, 4, 4]]]

sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list()                   # needs sage.combinat
[[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]

sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list(distinct=True)      # needs sage.combinat
[[[1, 2, 3], [4, 4, 4]],
 [[1, 3, 2], [4, 4, 4]],
 [[1, 2, 3], [4, 4, 4]],
 [[1, 3, 2], [4, 4, 4]]]

class Element#

Bases: 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]                                                       # needs sage.combinat
sage: elt.check()                                                       # needs sage.combinat

iterator(distinct=False)#

AUTHORS:

Robert Miller

EXAMPLES:

sage: CyclicPermutationsOfPartition([[1,2,3,4],        # indirect doctest   # needs sage.combinat
....:                                [5,6,7]]).list()
[[[1, 2, 3, 4], [5, 6, 7]],
 [[1, 2, 4, 3], [5, 6, 7]],
 [[1, 3, 2, 4], [5, 6, 7]],
 [[1, 3, 4, 2], [5, 6, 7]],
 [[1, 4, 2, 3], [5, 6, 7]],
 [[1, 4, 3, 2], [5, 6, 7]],
 [[1, 2, 3, 4], [5, 7, 6]],
 [[1, 2, 4, 3], [5, 7, 6]],
 [[1, 3, 2, 4], [5, 7, 6]],
 [[1, 3, 4, 2], [5, 7, 6]],
 [[1, 4, 2, 3], [5, 7, 6]],
 [[1, 4, 3, 2], [5, 7, 6]]]

sage: CyclicPermutationsOfPartition([[1,2,3,4],[4,4,4]]).list()             # needs sage.combinat
[[[1, 2, 3, 4], [4, 4, 4]],
 [[1, 2, 4, 3], [4, 4, 4]],
 [[1, 3, 2, 4], [4, 4, 4]],
 [[1, 3, 4, 2], [4, 4, 4]],
 [[1, 4, 2, 3], [4, 4, 4]],
 [[1, 4, 3, 2], [4, 4, 4]]]

sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list()               # needs sage.combinat
[[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]

sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list(distinct=True)  # needs sage.combinat
[[[1, 2, 3], [4, 4, 4]],
 [[1, 3, 2], [4, 4, 4]],
 [[1, 2, 3], [4, 4, 4]],
 [[1, 3, 2], [4, 4, 4]]]

list(distinct=False)#

EXAMPLES:

sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list()               # needs sage.combinat
[[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]
sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list(distinct=True)  # needs sage.combinat
[[[1, 2, 3], [4, 4, 4]],
 [[1, 3, 2], [4, 4, 4]],
 [[1, 2, 3], [4, 4, 4]],
 [[1, 3, 2], [4, 4, 4]]]

class sage.combinat.permutation.PatternAvoider(parent, patterns)#

Bases: GenericBacktracker

EXAMPLES:

sage: from sage.combinat.permutation import PatternAvoider
sage: P = Permutations(4)
sage: p = PatternAvoider(P, [[1,2,3]])
sage: loads(dumps(p))
<sage.combinat.permutation.PatternAvoider object at 0x...>

class sage.combinat.permutation.Permutation(parent, l, check=True)#

Bases: CombinatorialElement

A permutation.

Converts l to a permutation on \(\{1, 2, \ldots, n\}\).

INPUT:

l – Can be any one of the following:
- an instance of Permutation,
- list of integers, viewed as one-line permutation notation. The construction checks that you give an acceptable entry. To avoid the check, use the check option.
- string, expressing the permutation in cycle notation.
- list of tuples of integers, expressing the permutation in cycle notation.
- a PermutationGroupElement
- a pair of two standard tableaux of the same shape. This yields the permutation obtained from the pair using the inverse of the Robinson-Schensted algorithm.
check (boolean) – whether to check that input is correct. Slows
the function down, but ensures that nothing bad happens. This is set to True by default.

Warning

Since github issue #13742 the input is checked for correctness : it is not accepted unless it actually is a permutation on \(\{1, \ldots, n\}\). It means that some Permutation() objects cannot be created anymore without setting check=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 methods left_action_product() and right_action_product() for multiplying permutations (these methods don’t depend on the setting). See github issue #14885 for more details.

Note

The bruhat* methods refer to the strong Bruhat order. To use the weak Bruhat order, look under permutohedron*.

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 1-cycle to increase this:

sage: p2 = Permutation( '(4,5)(10)' ); p2
[1, 2, 3, 5, 4, 6, 7, 8, 9, 10]
sage: len(p); len(p2)
5
10

We construct a Permutation from a PermutationGroupElement:

sage: g = PermutationGroupElement([2,1,3])                                      # needs sage.groups
sage: Permutation(g)                                                            # needs sage.groups
[2, 1, 3]

From a pair of tableaux of the same shape. This uses the inverse of the Robinson-Schensted algorithm:

sage: # needs sage.combinat
sage: p = [[1, 4, 7], [2, 5], [3], [6]]
sage: q = [[1, 2, 5], [3, 6], [4], [7]]
sage: P = Tableau(p)
sage: Q = Tableau(q)
sage: Permutation( (p, q) )
[3, 6, 5, 2, 7, 4, 1]
sage: Permutation( [p, q] )
[3, 6, 5, 2, 7, 4, 1]
sage: Permutation( (P, Q) )
[3, 6, 5, 2, 7, 4, 1]
sage: Permutation( [P, Q] )
[3, 6, 5, 2, 7, 4, 1]

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()                                 # needs sage.combinat
[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()                              # needs sage.combinat
1

action(a)#

Return the action of the permutation self on a list a.

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 pattern patt.

EXAMPLES:

sage: Permutation([6,2,5,4,3,1]).avoids([4,2,3,1])                          # needs sage.combinat
False
sage: Permutation([6,1,2,5,4,3]).avoids([4,2,3,1])                          # needs sage.combinat
True
sage: Permutation([6,1,2,5,4,3]).avoids([3,4,1,2])                          # needs sage.combinat
True

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 one-line notation), and thus a binary search tree associated to a permutation is defined.

If the optional keyword variable left_to_right is set to False, the word \(w\) is being traversed from right to left instead.

EXAMPLES:

sage: Permutation([1,4,3,2]).binary_search_tree()                           # needs sage.graphs
1[., 4[3[2[., .], .], .]]
sage: Permutation([4,1,3,2]).binary_search_tree()                           # needs sage.graphs
4[1[., 3[2[., .], .]], .]

By passing the option left_to_right=False one can have the insertion going from right to left:

sage: Permutation([1,4,3,2]).binary_search_tree(False)                      # needs sage.graphs
2[1[., .], 3[., 4[., .]]]
sage: Permutation([4,1,3,2]).binary_search_tree(False)                      # needs sage.graphs
2[1[., .], 3[., 4[., .]]]

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()                     # needs sage.graphs
[., [[[., .], .], .]]
sage: Permutation([4,1,3,2]).binary_search_tree_shape()                     # needs sage.graphs
[[., [[., .], .]], .]

By passing the option left_to_right=False one can have the insertion going from right to left:

sage: Permutation([1,4,3,2]).binary_search_tree_shape(False)                # needs sage.graphs
[[., .], [., [., .]]]
sage: Permutation([4,1,3,2]).binary_search_tree_shape(False)                # needs sage.graphs
[[., .], [., [., .]]]

bruhat_greater()#

Return the combinatorial class of permutations greater than or equal to self in the Bruhat order (on the symmetric group containing self).

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 to self 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 to self 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 if self is less or equal to p2 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 and p2 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 containing self) such that there is no permutation between one of those and self.

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 containing self) such that there is no permutation between one of those and self.

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 containing self).

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 containing self) such that there is no permutation between one of those and self.

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 containing self) such that there is no permutation between one of those and self.

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 1-cycles in the string.

EXAMPLES:

sage: Permutation([1,2,3]).cycle_string()
'()'
sage: Permutation([2,1,3]).cycle_string()
'(1,2)'
sage: Permutation([2,3,1]).cycle_string()
'(1,2,3)'
sage: Permutation([2,1,3]).cycle_string(singletons=True)
'(1,2)(3)'

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 of self.

This is a non-increasing sequence of the cycle lengths of self.

EXAMPLES:

sage: Permutation([3,1,2,4]).cycle_type()                                   # needs sage.combinat
[3, 1]

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

runs()

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)] where i ranges over the descents of p.

A descent of a permutation p is an integer i such that p(i) > p(i+1).

REFERENCES:

[GS1984]

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 to True.

INPUT:

final_descent – boolean (default False); if True, the last position of a non-empty permutation is also considered as a descent
side – 'right' (default) or 'left'; if 'left', return the descents of the inverse permutation
positive – boolean (default False); if True, return the positions that are not descents
from_zero – boolean (default False); if True, return the positions starting from \(0\)
index_set – list (default: [1, ..., n-1] where self is a permutation of n); 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 \(i \in \{ 1, 2, \ldots, n-1 \}\) satisfying \(p(i) > p(i+1)\). The descent set of a composition \(c = (i_1, i_2, \ldots, i_k)\) is defined as the set \(\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}\).

EXAMPLES:

sage: Permutation([1,3,2,4]).descents_composition()
[2, 2]
sage: Permutation([4,1,6,7,2,3,8,5]).descents_composition()
[1, 3, 3, 1]
sage: Permutation([]).descents_composition()
[]

destandardize(weight, ordered_alphabet=None)#

Return destandardization of self with respect to weight and ordered_alphabet.

INPUT:

weight – list or tuple of nonnegative integers that sum to \(n\) if self 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 to self

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 word w over the specified ordered alphabet with evaluation weight such that w.standard_permutation() is self.

EXAMPLES:

sage: p = Permutation([1,2,5,3,6,4])
sage: p.destandardize([3,1,2])                                              # needs sage.combinat
word: 113132
sage: p = Permutation([2,1,3])
sage: p.destandardize([2,1])
Traceback (most recent call last):
...
ValueError: Standardization with weight [2, 1] is not possible!

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

\(1\),
\(|1|4 \to 14\),
\(|14|2 \to 412\),
\(|4|1|2|5 \to 4125\),
\(|4|125|3 \to 45123\).

So \(\phi([1,4,2,5,3]) = [4,5,1,2,3]\).

See section 2 of [FS1978], and the proof of Proposition 1.4.6 in [EnumComb1].

See also

foata_bijection_inverse() for the inverse map.

EXAMPLES:

sage: Permutation([1,2,4,3]).foata_bijection()
[4, 1, 2, 3]
sage: Permutation([2,5,1,3,4]).foata_bijection()
[2, 1, 3, 5, 4]

sage: P = Permutation([2,5,1,3,4])
sage: P.major_index() == P.foata_bijection().number_of_inversions()
True

sage: all( P.major_index() == P.foata_bijection().number_of_inversions()
....:      for P in Permutations(4) )
True

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 1-line notation.

See also

fundamental_transformation(), which is a similar map that is defined by instead taking \(a_{j,1} > a_{j,i}\) and is bijective.

EXAMPLES:

sage: P = Permutations(5)
sage: x = P([1, 5, 3, 4, 2])
sage: x.forget_cycles()
[1, 2, 5, 3, 4]

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)])                        # needs sage.combinat sage.graphs
sage: R.<q> = QQ[]
sage: sum(q^B.rank_function()(x) for x in B)                                # needs sage.combinat sage.graphs
q^5 + 2*q^4 + 3*q^3 + 3*q^2 + 2*q + 1

We check the statement in [CC2013] that the posets \(C_{[1,3,1,1]}\) and \(C_{[1,3,2]}\) are isomorphic:

sage: l2 = [p for p in P if [len(t) for t in p.to_cycles()] == [1,3,1,1]]
sage: l2 = [p.forget_cycles().inverse() for p in l2]
sage: B2 = Poset([l2, lambda x,y: x.bruhat_lequal(y)])                      # needs sage.combinat sage.graphs
sage: B.is_isomorphic(B2)                                                   # needs sage.combinat sage.graphs
True

See also

fundamental_transformation().

fundamental_transformation()#

Return the image of the permutation self under the Renyi-Foata-Schuetzenberger fundamental transformation.

The fundamental transformation is a bijection from the set of all permutations of \(\{1, 2, \ldots, n\}\) to itself, which transforms any such permutation \(w\) as follows: Write \(w\) in cycle form, with each cycle starting with its highest element, and the cycles being sorted in increasing order of their highest elements. Drop the parentheses in the resulting expression, thus reading it as a one-line notation of a new permutation \(u\). Then, \(u\) is the image of \(w\) under the fundamental transformation.

See [EnumComb1], Proposition 1.3.1.

See also

fundamental_transformation_inverse() for the inverse map.

forget_cycles() for a similar (but non-bijective) map where each cycle is starting from its lowest element.

EXAMPLES:

sage: Permutation([5, 1, 3, 4, 2]).fundamental_transformation()
[3, 4, 5, 2, 1]
sage: Permutations(5)([1, 5, 3, 4, 2]).fundamental_transformation()
[1, 3, 4, 5, 2]
sage: Permutation([8, 4, 7, 2, 9, 6, 5, 1, 3]).fundamental_transformation()
[4, 2, 6, 8, 1, 9, 3, 7, 5]

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 Renyi-Foata-Schuetzenberger fundamental transformation.

The inverse of the fundamental transformation is a bijection from the set of all permutations of \(\{1, 2, \ldots, n\}\) to itself, which transforms any such permutation \(w\) as follows: Let \(I = \{ i_1 < i_2 < \cdots < i_k \}\) be the set of all left-to-right maxima of \(w\) (that is, of all indices \(j\) such that \(w(j)\) is bigger than each of \(w(1), w(2), \ldots, w(j-1)\)). The image of \(w\) under the inverse of the fundamental transformation is the permutation \(u\) that sends \(w(i-1)\) to \(w(i)\) for all \(i \notin I\) (notice that this makes sense, since \(1 \in I\) whenever \(n > 0\)), while sending each \(w(i_p - 1)\) (with \(p \geq 2\)) to \(w(i_{p-1})\). Here, we set \(i_{k+1} = n+1\).

See [EnumComb1], Proposition 1.3.1.

See also

fundamental_transformation() for the inverse map.

EXAMPLES:

sage: Permutation([3, 4, 5, 2, 1]).fundamental_transformation_inverse()
[5, 1, 3, 4, 2]
sage: Permutation([4, 2, 6, 8, 1, 9, 3, 7, 5]).fundamental_transformation_inverse()
[8, 4, 7, 2, 9, 6, 5, 1, 3]

grade()#

Return the size of self.

EXAMPLES:

sage: Permutation([3,4,1,2,5]).size()
5

has_nth_root(n)#

Decide if self has n-th roots.

An n-th root of the permutation \(\sigma\) is a permutation \(\gamma\) such that \(\gamma^n = \sigma\).

Note that the number of n-th roots only depends on the cycle type of self.

EXAMPLES:

sage: sigma = Permutations(5).identity()
sage: sigma.has_nth_root(3)
True

sage: sigma = Permutation('(1, 3)')
sage: sigma.has_nth_root(2)
False

See also

nth_roots()
number_of_nth_roots()

has_pattern(patt)#

Test whether the permutation self contains the pattern patt.

EXAMPLES:

sage: Permutation([3,5,1,4,6,2]).has_pattern([1,3,2])                       # needs sage.combinat
True

hyperoctahedral_double_coset_type()#

Return the coset-type of self as a partition.

self must be a permutation of even size \(2n\). The coset-type determines the double class of the permutation, that is its image in \(H_n \backslash S_{2n} / H_n\), where \(H_n\) is the \(n\)-th hyperoctahedral group.

The coset-type is determined as follows. Consider the perfect matching \(\{\{1,2\},\{3,4\},\dots,\{2n-1,2n\}\}\) and its image by self, and draw them simultaneously as edges of a graph whose vertices are labeled by \(1,2,\dots,2n\). The coset-type is the ordered sequence of the semi-lengths of the cycles of this graph (see Chapter VII of [Mac1995] for more details, particularly Section VII.2).

EXAMPLES:

sage: # needs sage.combinat
sage: p = Permutation([3, 4, 6, 1, 5, 7, 2, 8])
sage: p.hyperoctahedral_double_coset_type()
[3, 1]
sage: all(p.hyperoctahedral_double_coset_type() ==
....:     p.inverse().hyperoctahedral_double_coset_type()
....:     for p in Permutations(4))
True
sage: Permutation([]).hyperoctahedral_double_coset_type()
[]
sage: Permutation([3,1,2]).hyperoctahedral_double_coset_type()
Traceback (most recent call last):
...
ValueError: [3, 1, 2] is a permutation of odd size and has no coset-type

idescents(final_descent=False, from_zero=False)#

Return a list of the idescents of self, that is the list of the descents of self’s inverse.

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 to True.

INPUT:

final_descent – boolean (default False); if True, the last position of a non-empty permutation is also considered as a descent
from_zero – optional boolean (default False); if False, 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 non-empty permutation is also considered as a descent.

EXAMPLES:

sage: Permutation([1,4,3,2]).idescents()
[2, 3]
sage: Permutation([1,4,3,2]).idescents_signature()
[1, -1, -1, 1]

imajor_index(final_descent=False)#

Return the inverse major index of the permutation self, which is the major index of the inverse of self.

The major index of a permutation \(p\) is the sum of the descents of \(p\). Since our permutation indices are 0-based, we need to add the number of descents.

With the final_descent option, the last position of a non-empty permutation is also considered as a descent.

EXAMPLES:

sage: Permutation([2,1,3]).imajor_index()
1
sage: Permutation([3,4,1,2]).imajor_index()
2
sage: Permutation([4,3,2,1]).imajor_index()
6

increasing_tree(compare=<built-in function min>)#

Return the increasing tree associated to self.

EXAMPLES:

sage: Permutation([1,4,3,2]).increasing_tree()                              # needs sage.graphs
1[., 2[3[4[., .], .], .]]
sage: Permutation([4,1,3,2]).increasing_tree()                              # needs sage.graphs
1[4[., .], 2[3[., .], .]]

By passing the option compare=max one can have the decreasing tree instead:

sage: Permutation([2,3,4,1]).increasing_tree(max)                           # needs sage.graphs
4[3[2[., .], .], 1[., .]]
sage: Permutation([2,3,1,4]).increasing_tree(max)                           # needs sage.graphs
4[3[2[., .], 1[., .]], .]

increasing_tree_shape(compare=<built-in function min>)#

Return the shape of the increasing tree associated with the permutation.

EXAMPLES:

sage: Permutation([1,4,3,2]).increasing_tree_shape()                        # needs sage.graphs
[., [[[., .], .], .]]
sage: Permutation([4,1,3,2]).increasing_tree_shape()                        # needs sage.graphs
[[., .], [[., .], .]]

By passing the option compare=max one can have the decreasing tree instead:

sage: Permutation([2,3,4,1]).increasing_tree_shape(max)                     # needs sage.graphs
[[[., .], .], [., .]]
sage: Permutation([2,3,1,4]).increasing_tree_shape(max)                     # needs sage.graphs
[[[., .], [., .]], .]

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 whether self is a derangement.

A permutation \(\sigma\) is a derangement if \(\sigma\) has no fixed points.

EXAMPLES:

sage: P = Permutation([1,4,2,3])
sage: P.is_derangement()
False
sage: P = Permutation([2,3,1])
sage: P.is_derangement()
True

is_even()#

Return True if the permutation self is even and False otherwise.

EXAMPLES:

sage: Permutation([1,2,3]).is_even()
True
sage: Permutation([2,1,3]).is_even()
False

is_simple()#

Return whether self is simple.

A permutation is simple if it does not send any proper sub-interval to a sub-interval.

For instance, [6,1,3,5,2,4] is not simple because it maps the interval [3,4,5,6] to [2,3,4,5], whereas [2,6,3,5,1,4] is simple.

See OEIS sequence A111111

EXAMPLES:

sage: g = Permutation([4,2,3,1])
sage: g.is_simple()
False

sage: g = Permutation([6,1,3,5,2,4])
sage: g.is_simple()
False

sage: g = Permutation([2,6,3,5,1,4])
sage: g.is_simple()
True

sage: [len([pi for pi in Permutations(n) if pi.is_simple()])
....:  for n in range(6)]
[1, 1, 2, 0, 2, 6]

ishift(i)#

Return the i-shift of self. If an i-shift of self can’t be performed, then self is returned.

An \(i\)-shift can be applied when \(i\) is not inbetween \(i-1\) and \(i+1\). The \(i\)-shift moves \(i\) to the other side, and leaves the relative positions of \(i-1\) and \(i+1\) in place. All other entries of the permutations are also left in place.

EXAMPLES:

Here, \(2\) is to the left of both \(1\) and \(3\). A \(2\)-shift can be applied which moves the \(2\) to the right and leaves \(1\) and \(3\) in their same relative order:

sage: Permutation([2,1,3]).ishift(2)
[1, 3, 2]

All entries other than \(i\), \(i-1\) 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 of self. If an i-switch of self can’t be performed, then self is returned.

An \(i\)-switch can be applied when the subsequence of self formed by the entries \(i-1\), \(i\) and \(i+1\) is neither increasing nor decreasing. In this case, this subsequence is reversed (i. e., its leftmost element and its rightmost element switch places), while all other letters of self 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 \(i-1\), \(i\) 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 with lp in such an order that lp is applied first and self is applied afterwards.

This is usually denoted by either self * lp or lp * 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 by self * 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 by lp * 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()                                 # needs sage.combinat
[[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])    # needs sage.combinat
....:     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.

A theorem of Schensted ([Sch1961]) states that an increasing subsequence of length \(i\) ends with the value entered in the \(i\)-th column of the p-tableau. The algorithm records which column of the p-tableau each value of the permutation is entered into, creates a digraph to record all increasing subsequences, and reads the paths from a source to a sink; these are the longest increasing subsequences.

EXAMPLES:

sage: Permutation([2,3,4,1]).longest_increasing_subsequences()              # needs sage.graphs
[[2, 3, 4]]
sage: Permutation([5, 7, 1, 2, 6, 4, 3]).longest_increasing_subsequences()  # needs sage.graphs
[[1, 2, 6], [1, 2, 4], [1, 2, 3]]

Note

This algorithm could be made faster using a balanced search tree for each column instead of sorted lists. See discussion on github issue #31451.

longest_increasing_subsequences_number()#

Return the number of increasing subsequences of maximal length in self.

The list of longest increasing subsequences of a permutation is given by longest_increasing_subsequences(), and the length of these subsequences is given by longest_increasing_subsequence_length().

The algorithm is similar to longest_increasing_subsequences(). Namely, the longest increasing subsequences are encoded as increasing sequences in a ranked poset from a smallest to a largest element. Their number can be obtained via dynamic programming: for each \(v\) in the poset we compute the number of paths from a smallest element to \(v\).

EXAMPLES:

sage: sum(p.longest_increasing_subsequences_number()
....:     for p in Permutations(8))
120770

sage: p = Permutations(50).random_element()
sage: (len(p.longest_increasing_subsequences()) ==                          # needs sage.graphs
....:  p.longest_increasing_subsequences_number())
True

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 0-based, we need to add the number of descents.

With the final_descent option, the last position of a non-empty permutation is also considered as a descent.

EXAMPLES:

sage: Permutation([2,1,3]).major_index()
1
sage: Permutation([3,4,1,2]).major_index()
2
sage: Permutation([4,3,2,1]).major_index()
6

multi_major_index(composition)#

Return the multimajor index of this permutation with respect to composition.

INPUT:

composition – a composition of the size() of this permutation

EXAMPLES:

sage: p = Permutation([5, 6, 2, 1, 3, 7, 4])
sage: p.multi_major_index([3, 2, 2])
[2, 0, 1]
sage: p.multi_major_index([7]) == [p.major_index()]
True
sage: p.multi_major_index([1]*7)
[0, 0, 0, 0, 0, 0, 0]
sage: Permutation([]).multi_major_index([])
[]

REFERENCES:

[JS2000]

next()#

Return the permutation that follows self in lexicographic order on the symmetric group containing self. If self is the last permutation, then next returns False.

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 in self.

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)
[]

nth_roots(n)#

Return all n-th roots of self (as a generator).

An n-th root of the permutation \(\sigma\) is a permutation \(\gamma\) such that \(\gamma^n = \sigma\).

Note that the number of n-th roots only depends on the cycle type of self.

EXAMPLES:

sage: sigma = Permutations(5).identity()
sage: list(sigma.nth_roots(3))
[[1, 4, 3, 5, 2], [1, 5, 3, 2, 4], [1, 2, 4, 5, 3], [1, 2, 5, 3, 4], [4, 2, 3, 5, 1], [5, 2, 3, 1, 4], [3, 2, 5, 4, 1],
 [5, 2, 1, 4, 3], [2, 5, 3, 4, 1], [5, 1, 3, 4, 2], [2, 3, 1, 4, 5], [3, 1, 2, 4, 5], [2, 4, 3, 1, 5], [4, 1, 3, 2, 5],
 [3, 2, 4, 1, 5], [4, 2, 1, 3, 5], [1, 3, 4, 2, 5], [1, 4, 2, 3, 5], [1, 3, 5, 4, 2], [1, 5, 2, 4, 3], [1, 2, 3, 4, 5]]

sage: sigma = Permutation('(1, 3)')
sage: list(sigma.nth_roots(2))
[]

For n >= 6, this algorithm begins to be more efficient than naive search (look at all permutations and test their n-th power).

See also

has_nth_root()
number_of_nth_roots()

number_of_descents(final_descent=False)#

Return the number of descents of self.

With the final_descent option, the last position of a non-empty permutation is also considered as a descent.

EXAMPLES:

sage: Permutation([1,4,3,2]).number_of_descents()
2
sage: Permutation([1,4,3,2]).number_of_descents(final_descent=True)
3

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 non-empty permutation is also considered as a descent.

EXAMPLES:

sage: Permutation([1,4,3,2]).number_of_idescents()
2
sage: Permutation([1,4,3,2]).number_of_idescents(final_descent=True)
3

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:

http://mathworld.wolfram.com/PermutationInversion.html

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 in self.

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)                                                    # needs sage.symbolic
sage: all( x.number_of_noninversions(2) == b - x.number_of_inversions()     # needs sage.symbolic
....:      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_nth_roots(n)#

Return the number of n-th roots of self.

An n-th root of the permutation \(\sigma\) is a permutation \(\gamma\) such that \(\gamma^n = \sigma\).

Note that the number of n-th roots only depends on the cycle type of self.

EXAMPLES:

sage: Sigma = Permutations(5).identity()
sage: Sigma.number_of_nth_roots(3)
21

sage: Sigma = Permutation('(1, 3)')
sage: Sigma.number_of_nth_roots(2)
0

See also

nth_roots()
has_nth_root()

number_of_peaks()#

Return the number of peaks of the permutation self.

A peak of a permutation \(p\) is an integer \(i\) such that \(p(i-1) < p(i)\) and \(p(i) > p(i+1)\).

EXAMPLES:

sage: Permutation([1,3,2,4,5]).number_of_peaks()
1
sage: Permutation([4,1,3,2,6,5]).number_of_peaks()
2

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_reduced_words()#

Return the number of reduced words of self without explicitly computing them all.

EXAMPLES:

sage: p = Permutation([6,4,2,5,1,8,3,7])
sage: len(p.reduced_words()) == p.number_of_reduced_words()                 # needs sage.combinat
True

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 permutation self.

EXAMPLES:

sage: Permutation([3,5,1,4,6,2]).pattern_positions([1,3,2])                 # needs sage.combinat
[[0, 1, 3], [2, 3, 5], [2, 4, 5]]

peaks()#

Return a list of the peaks of the permutation self.

A peak of a permutation \(p\) is an integer \(i\) such that \(p(i-1) < p(i)\) and \(p(i) > p(i+1)\).

EXAMPLES:

sage: Permutation([1,3,2,4,5]).peaks()
[1]
sage: Permutation([4,1,3,2,6,5]).peaks()
[2, 4]
sage: Permutation([]).peaks()
[]

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: # needs sage.combinat sage.graphs
sage: Permutation([3,1,5,4,2]).permutation_poset().cover_relations()
[[(2, 1), (5, 2)],
 [(2, 1), (3, 5)],
 [(2, 1), (4, 4)],
 [(1, 3), (3, 5)],
 [(1, 3), (4, 4)]]
sage: Permutation([]).permutation_poset().cover_relations()
[]
sage: Permutation([1,3,2]).permutation_poset().cover_relations()
[[(1, 1), (2, 3)], [(1, 1), (3, 2)]]
sage: Permutation([1,2]).permutation_poset().cover_relations()
[[(1, 1), (2, 2)]]
sage: P = Permutation([1,5,2,4,3])

This should hold for any \(P\):

sage: P.permutation_poset().greene_shape() == P.RS_partition()              # needs sage.combinat sage.graphs
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 and other in the right permutohedron order (or, if side 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 \(\psi\).

ALGORITHM:

It is enough to construct the join of any two permutations \(\pi\) and \(\psi\) in \(S_n\) with respect to the right weak order. (The join of \(\pi\) and \(\psi\) with respect to the left weak order is the inverse of the join of \(\pi^{-1}\) and \(\psi^{-1}\) with respect to the right weak order.) Start with an empty list \(l\) (denoted xs in the actual code). For \(i = 1, 2, \ldots, n\) (in this order), we insert \(i\) into this list in the rightmost possible position such that any letter in \(\{ 1, 2, ..., i-1 \}\) which appears further right than \(i\) in either \(\pi\) or \(\psi\) (or both) must appear further right than \(i\) in the resulting list. After all numbers are inserted, we are left with a list which is precisely the join of \(\pi\) and \(\psi\) (in one-line notation). This algorithm is due to Markowsky, [Mar1994] (Theorem 1 (a)).

AUTHORS:

Viviane Pons and Darij Grinberg, 18 June 2014.

EXAMPLES:

sage: p = Permutation([3,1,2])
sage: q = Permutation([1,3,2])
sage: p.permutohedron_join(q)
[3, 1, 2]
sage: r = Permutation([2,1,3])
sage: r.permutohedron_join(p)
[3, 2, 1]

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 if self is less or equal to p2 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 and other in the right permutohedron order (or, if side is set to 'left', in the left permutohedron order).

The permutohedron orders (see permutohedron_lequal()) are lattices; the meet operation refers to this lattice structure. It is connected to the join operation by the following simple symmetry property: If \(\pi\) and \(\psi\) are two permutations \(\pi\) and \(\psi\) in the symmetric group \(S_n\), and if \(w_0\) denotes the permutation \((n, n-1, \ldots, 1) \in S_n\), then

\[\pi \wedge \psi = w_0 \circ ((w_0 \circ \pi) \vee (w_0 \circ \psi)) = ((\pi \circ w_0) \vee (\psi \circ w_0)) \circ w_0\]

and

\[\pi \vee \psi = w_0 \circ ((w_0 \circ \pi) \wedge (w_0 \circ \psi)) = ((\pi \circ w_0) \wedge (\psi \circ w_0)) \circ w_0,\]

where \(\wedge\) means meet and \(\vee\) means join.

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 and self.

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 and self.

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 containing self. If self is the first permutation, then it returns False.

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 which self 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 well-known fact that every permutation can be written as a product of adjacent transpositions. In more detail: If \(n\) is a nonnegative integer, we can define the transpositions \(s_i = (i, i+1) \in S_n\) for all \(i \in \{ 1, 2, \ldots, n-1 \}\), and every \(p \in S_n\) can then be written as a product \(s_{i_1} s_{i_2} \cdots s_{i_k}\) for some sequence \((i_1, i_2, \ldots, i_k)\) of elements of \(\{ 1, 2, \ldots, n-1 \}\) (here \(\{ 1, 2, \ldots, n-1 \}\) denotes the empty set when \(n \leq 1\)). Fixing a \(p\), the sequences \((i_1, i_2, \ldots, i_k)\) of smallest length satisfying \(p = s_{i_1} s_{i_2} \cdots s_{i_k}\) are called the reduced words of \(p\). (Their length is the Coxeter length of \(p\), and can be computed using length().)

Note that the product of permutations is defined here in such a way that \((pq)(i) = p(q(i))\) for all permutations \(p\) and \(q\) and each \(i \in \{ 1, 2, \ldots, n \}\) (this is the same convention as in left_action_product(), but not the default semantics of the \(*\) operator on permutations in Sage). Thus, for instance, \(s_2 s_1\) is the permutation obtained by first transposing \(1\) with \(2\) and then transposing \(2\) with \(3\).

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.

However, return [1] rather than [] if self is the identity permutation.

This is mostly a helper method for sage.combinat.schubert_polynomial, where it is used to normalize finitary permutations of \(\{1,2,3,\ldots\}\).

EXAMPLES:

sage: Permutation([2,1,3]).remove_extra_fixed_points()
[2, 1]
sage: Permutation([1,2,3,4]).remove_extra_fixed_points()
[1]
sage: Permutation([2,1]).remove_extra_fixed_points()
[2, 1]
sage: Permutation([]).remove_extra_fixed_points()
[1]

See also

retract_plain()

retract_direct_product(m)#

Return the direct-product retract of the permutation self \(\in S_n\) to \(S_m\), where \(m \leq n\). If this retract is undefined, then None is returned.

If \(p \in S_n\) is a permutation, and \(m\) is a nonnegative integer less or equal to \(n\), then the direct-product retract of \(p\) to \(S_m\) is defined only if \(p([m]) = [m]\), where \([m]\) denotes the interval \(\{1, 2, \ldots, m\}\). In this case, it is defined as the permutation written \((p(1), p(2), \ldots, p(m))\) in one-line notation.

EXAMPLES:

sage: Permutation([4,1,2,3,5]).retract_direct_product(4)
[4, 1, 2, 3]
sage: Permutation([4,1,2,3,5]).retract_direct_product(3)

sage: Permutation([1,4,2,3,6,5]).retract_direct_product(5)
sage: Permutation([1,4,2,3,6,5]).retract_direct_product(4)
[1, 4, 2, 3]
sage: Permutation([1,4,2,3,6,5]).retract_direct_product(3)
sage: Permutation([1,4,2,3,6,5]).retract_direct_product(2)
sage: Permutation([1,4,2,3,6,5]).retract_direct_product(1)
[1]
sage: Permutation([1,4,2,3,6,5]).retract_direct_product(0)
[]

sage: all( p.retract_direct_product(3) == p for p in Permutations(3) )
True

retract_okounkov_vershik(m)#

Return the Okounkov-Vershik retract of the permutation self \(\in S_n\) to \(S_m\), where \(m \leq n\).

If \(p \in S_n\) is a permutation, and \(m\) is a nonnegative integer less or equal to \(n\), then the Okounkov-Vershik retract of \(p\) to \(S_m\) is defined as the permutation in \(S_m\) which sends every \(i \in \{1, 2, \ldots, m\}\) to \(p^{k_i}(i)\), where \(k_i\) is the smallest positive integer \(k\) satisfying \(p^k(i) \leq m\).

In other words, the Okounkov-Vershik retract of \(p\) is the permutation whose disjoint cycle decomposition is obtained by removing all letters strictly greater than \(m\) from the decomposition of \(p\) into disjoint cycles (and removing all cycles which are emptied in the process).

When \(m = n-1\), the Okounkov-Vershik retract (as a map \(S_n \to S_{n-1}\)) is the map \(\widetilde{p}_n\) introduced in Section 7 of [VO2005], and appears as (3.20) in [CST2010]. In the general case, the Okounkov-Vershik retract of a permutation in \(S_n\) to \(S_m\) can be obtained by first taking its Okounkov-Vershik retract to \(S_{n-1}\), then that of the resulting permutation to \(S_{n-2}\), etc. until arriving in \(S_m\).

EXAMPLES:

sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(4)
[4, 1, 2, 3]
sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(3)
[3, 1, 2]
sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(2)
[2, 1]
sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(1)
[1]
sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(0)
[]

sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(5)
[1, 4, 2, 3, 5]
sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(4)
[1, 4, 2, 3]
sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(3)
[1, 3, 2]
sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(2)
[1, 2]
sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(1)
[1]
sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(0)
[]

sage: Permutation([6,5,4,3,2,1]).retract_okounkov_vershik(5)
[1, 5, 4, 3, 2]
sage: Permutation([6,5,4,3,2,1]).retract_okounkov_vershik(4)
[1, 2, 4, 3]

sage: Permutation([1,5,2,6,3,7,4,8]).retract_okounkov_vershik(4)
[1, 3, 2, 4]

sage: all( p.retract_direct_product(3) == p for p in Permutations(3) )
True

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, then None is returned.

If \(p \in S_n\) is a permutation, and \(m\) is a nonnegative integer less or equal to \(n\), then the plain retract of \(p\) to \(S_m\) is defined only if every \(i > m\) satisfies \(p(i) = i\). In this case, it is defined as the permutation written \((p(1), p(2), \ldots, p(m))\) in one-line notation.

EXAMPLES:

sage: Permutation([4,1,2,3,5]).retract_plain(4)
[4, 1, 2, 3]
sage: Permutation([4,1,2,3,5]).retract_plain(3)

sage: Permutation([1,3,2,4,5,6]).retract_plain(3)
[1, 3, 2]
sage: Permutation([1,3,2,4,5,6]).retract_plain(2)

sage: Permutation([1,2,3,4,5]).retract_plain(1)
[1]
sage: Permutation([1,2,3,4,5]).retract_plain(0)
[]

sage: all( p.retract_plain(3) == p for p in Permutations(3) )
True

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 with rp in such an order that self is applied first and rp is applied afterwards.

This is usually denoted by either self * rp or rp * 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 by rp * 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 by self * 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 and other the maximal element.

See permutohedron_lequal() for the definition of the permutohedron orders.

EXAMPLES:

sage: p = Permutation([2, 1, 4, 5, 3]); q = Permutation([2, 5, 4, 1, 3])
sage: p.right_permutohedron_interval(q)                                     # needs sage.graphs sage.modules
[[2, 4, 5, 1, 3], [2, 4, 1, 5, 3], [2, 1, 4, 5, 3],
 [2, 1, 5, 4, 3], [2, 5, 1, 4, 3], [2, 5, 4, 1, 3]]

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 and other the maximal element.

See permutohedron_lequal() for the definition of the permutohedron orders.

EXAMPLES:

sage: p = Permutation([2, 1, 4, 5, 3]); q = Permutation([2, 5, 4, 1, 3])
sage: p.right_permutohedron_interval(q)  # indirect doctest                 # needs sage.graphs sage.modules
[[2, 4, 5, 1, 3], [2, 4, 1, 5, 3], [2, 1, 4, 5, 3],
 [2, 1, 5, 4, 3], [2, 5, 1, 4, 3], [2, 5, 4, 1, 3]]

right_tableau()#

Return the right standard tableau after performing the RSK algorithm on self.

EXAMPLES:

sage: Permutation([1,4,3,2]).right_tableau()                                # needs sage.combinat
[[1, 2], [3], [4]]

robinson_schensted()#

Return the pair of standard tableaux obtained by running the Robinson-Schensted algorithm on self.

This can also be done by running RSK() on self (with the optional argument check_standard=True to return standard Young tableaux).

EXAMPLES:

sage: Permutation([6,2,3,1,7,5,4]).robinson_schensted()                     # needs sage.combinat
[[[1, 3, 4], [2, 5], [6, 7]], [[1, 3, 5], [2, 6], [4, 7]]]

rothe_diagram()#

Return the Rothe diagram of self.

EXAMPLES:

sage: p = Permutation([4,2,1,3])
sage: D = p.rothe_diagram(); D                                              # needs sage.combinat
[(0, 0), (0, 1), (0, 2), (1, 0)]
sage: D.pp()                                                                # needs sage.combinat
O O O .
O . . .
. . . .
. . . .

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:

http://mathworld.wolfram.com/PermutationRun.html

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 permutation other. These operations are also known as the Loday-Ronco over and under operations.

INPUT:

other – a permutation, a list, a tuple, or any iterable representing a permutation.
side – (default: "right") the string “left” or “right”.

OUTPUT:

If side is "right", the method returns the permutation obtained by concatenating self with the letters of other incremented by the size of self. This is what is called side / other in [LR0102066], and denoted as the “over” operation. Otherwise, i. e., when side is "left", the method returns the permutation obtained by concatenating the letters of other incremented by the size of self with self. This is what is called side \ 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 and other.

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 and other.

EXAMPLES:

sage: # needs sage.graphs sage.modules
sage: Permutation([]).shifted_shuffle(Permutation([]))
[[]]
sage: Permutation([1, 2, 3]).shifted_shuffle(Permutation([1]))
[[4, 1, 2, 3], [1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3]]
sage: Permutation([1, 2]).shifted_shuffle(Permutation([2, 1]))
[[4, 1, 3, 2], [4, 3, 1, 2], [1, 4, 3, 2],
 [1, 4, 2, 3], [1, 2, 4, 3], [4, 1, 2, 3]]
sage: Permutation([1]).shifted_shuffle([1])
[[2, 1], [1, 2]]
sage: p = Permutation([3, 1, 5, 4, 2])
sage: len(p.shifted_shuffle(Permutation([2, 1, 4, 3])))
126

The shifted shuffle product is associative. We can test this on an admittedly toy example:

sage: all( all( all( sorted(flatten([abs.shifted_shuffle(c)                 # needs sage.graphs sage.modules
....:                                for abs in a.shifted_shuffle(b)]))
....:                == sorted(flatten([a.shifted_shuffle(bcs)
....:                                   for bcs in b.shifted_shuffle(c)]))
....:                for c in Permutations(2) )
....:           for b in Permutations(2) )
....:      for a in Permutations(2) )
True

The shifted_shuffle method on permutations gives the same permutations as the shifted_shuffle method on words (but is faster):

sage: all( all( sorted(p1.shifted_shuffle(p2))                              # needs sage.combinat sage.graphs sage.modules sage.rings.finite_rings
....:           == sorted([Permutation(p) for p in
....:                      Word(p1).shifted_shuffle(Word(p2))])
....:           for p2 in Permutations(3) )
....:      for p1 in Permutations(2) )
True

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").
- "chord-diagram" – the permutation is displayed as a directed graph, all of its vertices being located on a circle.

All additional arguments are forwarded to the show subcalls.

EXAMPLES:

sage: P20 = Permutations(20)
sage: P20.random_element().show(representation="cycles")                    # needs sage.graphs sage.plot
sage: P20.random_element().show(representation="chord-diagram")             # needs sage.graphs sage.plot
sage: P20.random_element().show(representation="braid")                     # needs sage.plot
sage: P20.random_element().show(representation="braid",                     # needs sage.plot
....:                           orientation='portrait')

sign()#

Return the signature of the permutation self. This is \((-1)^l\), where \(l\) is the number of inversions of self.

Note

sign() can be used as an alias for signature().

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 of self.

Note

sign() can be used as an alias for signature().

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 Simion-Schmidt map which sends an arbitrary permutation to a pattern avoiding permutation, where the permutation pattern is one of four length-three patterns. This method also implements the bijection between (for example) [1,2,3]- and [1,3,2]-avoiding permutations.

INPUT:

avoid – one of the patterns [1,2,3], [1,3,2], [3,1,2], [3,2,1].

EXAMPLES:

sage: P = Permutations(6)
sage: p = P([4,5,1,6,3,2])
sage: pl = [ [1,2,3], [1,3,2], [3,1,2], [3,2,1] ]
sage: for q in pl:                                                          # needs sage.combinat
....:     s = p.simion_schmidt(q)
....:     print("{} {}".format(s, s.has_pattern(q)))
[4, 6, 1, 5, 3, 2] False
[4, 2, 1, 3, 5, 6] False
[4, 5, 3, 6, 2, 1] False
[4, 5, 1, 6, 2, 3] False

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 stack-sortable.

See Wikipedia article Stack-sortable_permutation

EXAMPLES:

sage: p = Permutation([2,1,5,3,4,9,7,8,6])
sage: p.stack_sort()
[1, 2, 3, 4, 5, 7, 6, 8, 9]

sage: S5 = Permutations(5)
sage: len([1 for s in S5 if s.stack_sort() == S5.one()])
42

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 one-line notation.) This definition comes from [HNT2005], Definition 8, where it is more generally applied to arbitrary words.

The equivalence class of a permutation \(p \in S_n\) under sylvester congruence is called the sylvester class of \(p\). It is an interval in the right permutohedron order (see permutohedron_lequal()) on \(S_n\).

This is related to the sylvester_class() method in that the equivalence class of a permutation \(\pi\) under sylvester congruence is the sylvester class of the right-to-left binary search tree of \(\pi\). However, the present method yields permutations, while the method on labelled binary trees yields plain lists.

If the variable left_to_right is set to True, the method instead iterates over the equivalence class of self with respect to the left sylvester congruence. The left sylvester congruence is easiest to define by saying that two permutations are equivalent under it if and only if their reverses (reverse()) are equivalent under (standard) sylvester congruence.

EXAMPLES:

The sylvester class of a permutation in \(S_5\):

sage: p = Permutation([3, 5, 1, 2, 4])
sage: sorted(p.sylvester_class())                                           # needs sage.combinat sage.graphs
[[1, 3, 2, 5, 4],
 [1, 3, 5, 2, 4],
 [1, 5, 3, 2, 4],
 [3, 1, 2, 5, 4],
 [3, 1, 5, 2, 4],
 [3, 5, 1, 2, 4],
 [5, 1, 3, 2, 4],
 [5, 3, 1, 2, 4]]

The sylvester class of a permutation \(p\) contains \(p\):

sage: all(p in p.sylvester_class() for p in Permutations(4))                # needs sage.combinat sage.graphs
True

Small cases:

sage: list(Permutation([]).sylvester_class())                               # needs sage.combinat sage.graphs
[[]]

sage: list(Permutation([1]).sylvester_class())                              # needs sage.combinat sage.graphs
[[1]]

The sylvester classes in \(S_3\):

sage: [sorted(p.sylvester_class()) for p in Permutations(3)]                # needs sage.combinat sage.graphs
[[[1, 2, 3]],
 [[1, 3, 2], [3, 1, 2]],
 [[2, 1, 3]],
 [[2, 3, 1]],
 [[1, 3, 2], [3, 1, 2]],
 [[3, 2, 1]]]

The left sylvester classes in \(S_3\):

sage: [sorted(p.sylvester_class(left_to_right=True))                        # needs sage.combinat sage.graphs
....:  for p in Permutations(3)]
[[[1, 2, 3]],
 [[1, 3, 2]],
 [[2, 1, 3], [2, 3, 1]],
 [[2, 1, 3], [2, 3, 1]],
 [[3, 1, 2]],
 [[3, 2, 1]]]

A left sylvester class in \(S_5\):

sage: p = Permutation([4, 2, 1, 5, 3])
sage: sorted(p.sylvester_class(left_to_right=True))                         # needs sage.combinat sage.graphs
[[4, 2, 1, 3, 5],
 [4, 2, 1, 5, 3],
 [4, 2, 3, 1, 5],
 [4, 2, 3, 5, 1],
 [4, 2, 5, 1, 3],
 [4, 2, 5, 3, 1],
 [4, 5, 2, 1, 3],
 [4, 5, 2, 3, 1]]

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              # needs sage.combinat sage.modules
[1 0 0]
[0 1 0]
[0 0 1]
sage: parent(m)                                                             # needs sage.combinat sage.modules
Alternating sign matrices of size 3

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()                               # needs sage.graphs
sage: d.edges(sort=True, labels=False)                                      # needs sage.graphs
[(1, 3), (2, 1), (3, 2)]
sage: P = Permutations(range(1, 10))
sage: d = Permutation(P.random_element()).to_digraph()                      # needs sage.graphs
sage: all(c.is_cycle()                                                      # needs sage.graphs
....:     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 \((m_1-m_2, m_2-m_3, \ldots, m_n)\), 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 non-empty permutation is also considered as a descent. This has an effect on the computation of major indices.

REFERENCES:

Carlitz, L. q-Bernoulli and Eulerian Numbers. Trans. Amer. Math. Soc. 76 (1954) 332-350. http://www.ams.org/journals/tran/1954-076-02/S0002-9947-1954-0060538-2/
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()                                      # needs sage.modules
[1 0 0]
[0 1 0]
[0 0 1]

Alternatively:

sage: matrix(Permutation([1,3,2]))                                          # needs sage.modules
[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()                                                     # needs sage.modules
[0 0 1]
[0 1 0]
[1 0 0]
sage: p.to_matrix()*q.to_matrix()                                           # needs sage.modules
[0 0 1]
[0 1 0]
[1 0 0]
sage: Permutations.options.mult='l2r'
sage: (p*q).to_matrix()                                                     # needs sage.modules
[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()                 # needs sage.groups
(1,2)(3,4)
sage: Permutation([1,2,3]).to_permutation_group_element()                   # needs sage.groups
()

to_tableau_by_shape(shape)#

Return a tableau of shape shape with the entries in self. 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) is self.

EXAMPLES:

sage: T = Permutation([3,4,1,2,5]).to_tableau_by_shape([3,2]); T            # needs sage.combinat
[[1, 2, 5], [3, 4]]
sage: T.reading_word_permutation()                                          # needs sage.combinat
[3, 4, 1, 2, 5]

weak_excedences()#

Return all the numbers self[i] such that self[i] >= i+1.

EXAMPLES:

sage: Permutation([1,4,3,2,5]).weak_excedences()
[1, 4, 3, 5]

class sage.combinat.permutation.Permutations#

Bases: UniqueRepresentation, Parent

Permutations.

Permutations(n) returns the class of permutations of n, if n is an integer, list, set, or string.

Permutations(n, k) returns the class of length-k partial permutations of n (where n 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 or k along with a keyword.

Permutations(descents=(list,n)) returns the class of permutations of \(n\) with descents in the positions specified by list. This uses the slightly nonstandard convention that the images of \(1,2,...,n\) under the permutation are regarded as positions \(0,1,...,n-1\), so for example the presence of \(1\) in list signifies that the permutations \(\pi\) should satisfy \(\pi(2) > \pi(3)\). Note that list 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 composition list.

Permutations(bruhat_smaller=p) and Permutations(bruhat_greater=p) return the class of permutations smaller-or-equal or greater-or-equal, respectively, than the given permutation p in the Bruhat order. (The Bruhat order is defined in bruhat_lequal(). It is also referred to as the strong Bruhat order.)

Permutations(recoils=p) returns the class of permutations whose recoils composition is p. Unlike the descents=(list, n) syntax, this actually takes a composition as input.

Permutations(recoils_fatter=p) and Permutations(recoils_finer=p) return the class of permutations whose recoils composition is fatter or finer, respectively, than the given composition p.

Permutations(n, avoiding=P) returns the class of permutations of n avoiding P. Here P may be a single permutation or a list of permutations; the returned class will avoid all patterns in P.

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 multi-set [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 multi-set [1, 1, 2] of length 2
sage: p.list()                                                                  # needs sage.libs.gap
[[1, 1], [1, 2], [2, 1]]

sage: p = Permutations(descents=([1], 4)); p
Standard permutations of 4 with descents [1]
sage: p.list()                                                                  # needs sage.graphs sage.modules
[[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()                                                                  # needs sage.graphs sage.modules
[[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()                                                                  # needs sage.graphs sage.modules
[[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()                                                                  # needs sage.graphs sage.modules
[[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()                                                           # needs sage.combinat
88
sage: p.random_element().parent() is p                                          # needs sage.combinat
True

Element#: alias of Permutation

options = Current options for Permutations - display: list - generator_name: s - latex: list - latex_empty_str: 1 - mult: l2r#

class sage.combinat.permutation.PermutationsNK(s, k)#

Bases: Permutations_setk

This exists solely for unpickling PermutationsNK objects created with Sage <= 6.3.

class sage.combinat.permutation.Permutations_mset(mset)#

Bases: 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 multi-set [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: # needs sage.modules
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 multi-set [(1, 0), (1, 1)]
sage: sorted(P)
[[(1, 0), (1, 1)], [(1, 1), (1, 0)]]

class Element#

Bases: 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 between 0 and self.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: 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()                                # needs sage.libs.gap
3

class sage.combinat.permutation.Permutations_nk(n, k)#

Bases: Permutations

Length-\(k\) partial permutations of \(\{1, 2, \ldots, n\}\).

class Element#

Bases: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: StandardPermutations_n_abstract

Generic class for subset of permutations avoiding a set of patterns.

property a#: self.a is deprecated; use patterns() instead.

cardinality()#

Return the cardinality of self.

EXAMPLES:

sage: P = Permutations(3, avoiding=[[2, 1, 3],[1,2,3]])
sage: P.cardinality()                                                       # needs sage.combinat
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: 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: 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: 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: 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=True)#

Bases: Permutation

apply_simple_reflection_left(i)#

Return self multiplied by the simple reflection s[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 reflection s[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 of self.

A left descent of a permutation \(\pi \in S_n\) means an index \(i \in \{ 1, 2, \ldots, n-1 \}\) such that \(s_i \circ \pi\) has smaller length than \(\pi\). Thus, a left descent of \(\pi\) is an index \(i \in \{ 1, 2, \ldots, n-1 \}\) satisfying \(\pi^{-1}(i) > \pi^{-1}(i+1)\).

Warning

The methods descents() and idescents() behave differently than their Weyl group counterparts. In particular, the indexing is 0-based. 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 of self.

A right descent of a permutation \(\pi \in S_n\) means an index \(i \in \{ 1, 2, \ldots, n-1 \}\) such that \(\pi \circ s_i\) has smaller length than \(\pi\). Thus, a right descent of \(\pi\) is an index \(i \in \{ 1, 2, \ldots, n-1 \}\) satisfying \(\pi(i) > \pi(i+1)\).

Warning

This ignores the multiplication convention in order to be consistent with other Coxeter operations in permutations (e.g., computing reduced_word()).

EXAMPLES:

sage: 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 ring
category – a category (default: the category of self)

EXAMPLES:

sage: # needs sage.groups sage.modules
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(); PG                                     # needs sage.groups
Symmetric group of order 4! as a permutation group

sage: G = SymmetricGroup(4)                                                 # needs sage.groups
sage: PG is G                                                               # needs sage.groups
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 \(A_{n-1}\).

EXAMPLES:

sage: A = SymmetricGroup([2,3,7]); A.cartan_type()                          # needs sage.combinat sage.groups
['A', 2]
sage: A = SymmetricGroup([]); A.cartan_type()                               # needs sage.combinat sage.groups
['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 in self.

INPUT:

g – a partition or an element of self

EXAMPLES:

sage: G = Permutations(5)
sage: g = G([2,3,4,1,5])
sage: G.conjugacy_class(g)                                                  # needs sage.combinat sage.graphs sage.groups
Conjugacy class of cycle type [4, 1] in Standard permutations of 5
sage: G.conjugacy_class(Partition([2, 1, 1, 1]))                            # needs sage.combinat sage.graphs sage.groups
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()                                                 # needs sage.combinat sage.graphs sage.groups
[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()         # needs sage.combinat sage.graphs sage.groups
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()                                 # needs sage.combinat sage.libs.flint
[[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 in self, then some fixed points are added.

EXAMPLES:

sage: PP = Permutations(5)
sage: PP.element_in_conjugacy_classes([2,2])                                # needs sage.combinat
[2, 1, 4, 3, 5]
sage: PP.element_in_conjugacy_classes([5, 5])                               # needs sage.combinat
Traceback (most recent call last):
...
ValueError: the size of the partition (=10) should be at most the size of the permutations (=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 \(\{ 1, 2, \ldots, n-1 \}\), 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 or p depending on input.

If a permutation p is given, return the rank of p in self. 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 \(\{ 1, 2, \ldots, n-1 \}\), where self 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: 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: Permutations

Permutations of \(\{1, \ldots, n\}\) with a fixed recoils composition.

class sage.combinat.permutation.StandardPermutations_recoilsfatter(recoils)#: Bases: Permutations

class sage.combinat.permutation.StandardPermutations_recoilsfiner(recoils)#: Bases: Permutations

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 Birkhoff-von 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 non-bistochastic 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 matrix
check (boolean) – set to True (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), where permutation is a Sage Permutation instance, and coeff its corresponding coefficient from the result of this function by applying the list function.
If you are interested in the matrix corresponding to a Permutation you will be glad to learn about the Permutation.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 the Matrix 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: # needs networkx sage.graphs sage.modules
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: # needs sage.modules
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]])                                    # needs sage.modules
sage: bounded_affine_permutation(A)                                             # needs sage.libs.flint sage.modules
[5, 6, 3, 4]

sage: A = Matrix(ZZ, [[0,1,0,1,0], [0,0,1,1,0]])                                # needs sage.modules
sage: bounded_affine_permutation(A)                                             # needs sage.libs.flint sage.modules
[1, 4, 7, 8, 5]

REFERENCES:

[KLS2013]

sage.combinat.permutation.bruhat_lequal(p1, p2)#

Return True if p1 is less than p2 in the Bruhat order.

Algorithm from mupad-combinat.

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])                            # needs sage.graphs sage.modules
[[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 github issue #13742):

sage: Permutation("(-12,2)(3,4)")
Traceback (most recent call last):
...
ValueError: all elements should be strictly positive integers, but I found -12
sage: Permutation("(1,2)(2,4)")
Traceback (most recent call last):
...
ValueError: the element 2 appears more than once in the input
sage: permutation.from_cycles(4, [[1,18]])
Traceback (most recent call last):
...
ValueError: you claimed that this is 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_cocode(lehmer, parent=Standard permutations)#

Return the permutation with Lehmer cocode lehmer.

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: import sage.combinat.permutation as permutation
sage: lcc = Permutation([2,1,5,4,3]).to_lehmer_cocode(); lcc
[0, 1, 0, 1, 2]
sage: permutation.from_lehmer_cocode(lcc)
[2, 1, 5, 4, 3]

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 the Permutation() 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 a PermutationGroupElement pge.

EXAMPLES:

sage: import sage.combinat.permutation as permutation
sage: pge = PermutationGroupElement([(1,2),(3,4)])                              # needs sage.groups
sage: permutation.from_permutation_group_element(pge)                           # needs sage.groups
[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 of rank. 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 if p1 is less than or equal to p2 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 iterable
key – (optional) a comparison key for the element x of p

EXAMPLES:

sage: # needs sage.combinat
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]