PermutationTools

Pattern avoidance, Foata and related maps, Bruhat and weak order, families of permutations.

Load with Needs["PermutationTools`"].

Background on symmetricfunctions.com: Permutations, Permutation patterns, Families of permutations.

Functions and symbols

AlternatingPermutations

AlternatingPermutations[n] returns a list of alternating permutations (up-down permutations), see A000111.

BruhatLowerOrderIdeal

BruhatLowerOrderIdeal[permutation] returns all permutations below the given permutation in strong Bruhat order.

CanonPermutations

CanonPermutations[lam] generates canon permutations with content lambda.

CarlitzMap

CarlitzMap[pi] sends inv to maj.

FoataMap

FoataMap[pi] sends maj to inv.

FromSubExcedance

FromSubExcedance[f] returns the permutation associated with a sub-excedance word f. FromSubExcedance[f, pi] applies the word to the initial permutation pi.

GeneratePAPS

GeneratePAPS[n] returns all parity-alternating permutations of size n. GeneratePAPS[n, condition] restricts both component permutations and the result using condition.

GenerateRAPS

GenerateRAPS[n, k, r] returns the restricted alternating permutations generated from k component permutations; r defaults to 1.

GrassmannPermutations

GrassmannPermutations[n] returns all permutations with at most one descent. See A000325.

Is123AvoidingQ

Is123AvoidingQ[permutation] returns True if the permutation avoids the pattern 123.

Background: Permutation patterns

Is132AvoidingQ

Is132AvoidingQ[permutation] returns True if the permutation avoids the pattern 132.

Background: Permutation patterns

Is213AvoidingQ

Is213AvoidingQ[permutation] returns True if the permutation avoids the pattern 213.

Background: Permutation patterns

Is231AvoidingQ

Is231AvoidingQ[permutation] returns True if the permutation avoids the pattern 231.

Background: Permutation patterns

Is312AvoidingQ

Is312AvoidingQ[permutation] returns True if the permutation avoids the pattern 312.

Background: Permutation patterns

Is321AvoidingQ

Is321AvoidingQ[permutation] returns True if the permutation avoids the pattern 321.

Background: Permutation patterns

IsPAPQ

IsPAPQ[permutation] returns True if odd positions contain odd values and even positions contain even values.

IsPermutationAvoidingQ

IsPermutationAvoidingQ[sigma,pi] returns true if pi avoids the pattern sigma.

Background: Permutation patterns

NQueensPermutations

NQueensPermutations[n] returns the permutations solving the n-queens problem on an n by n board.

PairToPAP

PairToPAP[{p1, p2}] interleaves p1 and p2 after mapping their entries to odd and even values, respectively.

PAPS123

PAPS123[n] returns all parity-alternating permutations of size n that avoid 123.

PAPS132

PAPS132[n] returns all parity-alternating permutations of size n that avoid 132.

PAPS213

PAPS213[n] returns all parity-alternating permutations of size n that avoid 213.

PAPS231

PAPS231[n] returns all parity-alternating permutations of size n that avoid 231.

PAPS312

PAPS312[n] returns all parity-alternating permutations of size n that avoid 312.

PAPS321

PAPS321[n] returns all parity-alternating permutations of size n that avoid 321.

PAPToPair

PAPToPair[pap] splits a parity-alternating permutation into its odd-position and even-position subsequences, undoing PairToPAP.

PermutationAllCycles

PermutationAllCycles[pi] returns all cycles of the permutation, including fixed-points.

PermutationCharge

PermutationCharge[permutation] returns the charge statistic of a permutation.

PermutationCocharge

PermutationCocharge[permutation] returns the cocharge statistic of a permutation.

PermutationCode

PermutationCode[p] returns the code of the permutation.

PermutationCycleMap

PermutationCycleMap[p] is the map defined on p.23 Stanley’s EC1, where one writes the permutation in cycle form, and removes parenthesises.

PermutationFromWord

PermutationFromWord[word, n] returns the permutation of [n] represented by a word of simple transpositions.

PermutationGenus

PermutationGenus[pi] returns the genus of the permutation.

PermutationMatrixPlot

PermutationMatrixPlot[pi] returns a graphical representation of the permutation.

PermutationSkewSum

PermutationSkewSum[p1,p2,…] returns the skew sum of the permutations (places the permutation matrices along the anti-diagonal of a block matrix).

PermutationSum

PermutationSum[p1,p2,…] returns the direct sum of the permutations (places the permutation matrices along the main diagonal of a block matrix).

PermutationType

PermutationType[pi] returns the partition of cycle lengths

ReducedWord

ReducedWord[pi] returns a reduced word for pi.

SeparablePermutationQ

SeparablePermutationQ[pi] returns true if the permutation is separable.

Si

Si[pi, i] swaps entries i and i+1 in a permutation pi. Si[i][pi] is the equivalent curried form.

SimionSchmidtMap

SimionSchmidtMap[perm] ,see https://core.ac.uk/download/pdf/82232421.pdf

SimsunPermutations

SimsunPermutations[n] returns all Simsun permutations.

SkewMergedPermutations

SkewMergedPermutations[n] returns the permutations in S_n avoiding 2143 and 3412.

SplitSeparablePermutation

SplitSeparablePermutation[pi] splits the permutation into smaller blocks. The original permutation is then a sum or skew sum of these blocks. Use option ‘False’ to only consider direct sum.

StirlingPermutations

StirlingPermutations[n] returns all Stirling permutations of order n.

StrongBruhatDownSet

StrongBruhatDownSet[permutation] returns all permutations below the given permutation in strong Bruhat order.

StrongBruhatGreaterQ

StrongBruhatGreaterQ[p1,p2] returns true iff p1
is greater-equal than p2 in strong Bruhat order. The identity is smaller than all other, and w0 is largest.

ToSubExcedance

ToSubExcedance[pi] returns the sub-excedance function associated with the permutation pi.

TupleToRAP

TupleToRAP[lists, r] interleaves a list of permutations into a restricted alternating permutation; r defaults to 1.

TypeBPermutations

TypeBPermutations[n] returns all permutations of type B.

WachsPermutations

WachsPermutations[n] returns a list of all Wachs permutations in S_n. See arxiv:2212.04932.

WeakLowerOrderIdeal

WeakLowerOrderIdeal[pi] returns all permutations below pi in the weak order

WeakOrderGreaterQ

WeakOrderGreaterQ[p1,p2] returns true iff p1 is greater
than p2 in weak order. The identity is smaller than all other, and w0 is largest.