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Permutation -- the Permutation type

Description

Permutations are constructed from lists. To create a permutation, use the permutation method.

i1 : p = permutation {3,1,2,4,5}

o1 = Permutation{3, 1, 2, 4, 5}

o1 : Permutation

Permutations must be constructed from lists consisting of only the integers $1 \dots n$. If a list contains any other elements or does not consist of the entire range, then an error is thrown.

Functions and methods returning a permutation:

  • extend(Permutation,ZZ) -- see extend -- rewrites a permutation as a permutation on more symbols
  • foataBijection(Permutation) -- see foataBijection -- computes the image of a permutation under the Foata bijection
  • inverse(Permutation) -- computes the inverse of a permutation
  • permutation(VisibleList)
  • Permutation * Permutation -- computes the product of two permutations
  • Permutation ^ ZZ -- computes the power of a permutation
  • randomPermutation(ZZ) -- see randomPermutation -- generates a random permutation
  • transposition(ZZ) -- see transposition -- constructs a transposition
  • transposition(ZZ,ZZ) -- see transposition -- constructs a transposition
  • trim(Permutation) -- see trim -- rewrites a permutation in its smallest representation

Methods that use a permutation:

  • ascendingRuns(Permutation) -- see ascendingRuns -- computes the ascending runs of a permutation
  • ascents(Permutation) -- see ascents -- computes the ascents of a permutation
  • avoidsPattern(Permutation,List) -- see avoidsPattern -- whether a permutation avoids a pattern
  • avoidsPatterns(Permutation,List) -- see avoidsPatterns -- whether a permutation simultaneously avoids a list of patterns
  • cycleDecomposition(Permutation) -- see cycleDecomposition -- computes the decomposition of a permutation as a product of disjoint cycles
  • cycleType(Permutation) -- see cycleType -- computes the cycle type of a permutation
  • descendingRuns(Permutation) -- see descendingRuns -- computes the descending runs of a permutation
  • descents(Permutation) -- see descents -- computes the descents of a permutation
  • exceedances(Permutation) -- see exceedances -- computes the exceedances of a permutation
  • expression(Permutation) (missing documentation)
  • extend(Permutation,Permutation) -- rewrites two permutations to be permutations on the same number of symbols
  • fixedPoints(Permutation) -- see fixedPoints -- computes the fixed points of a permutation
  • html(Permutation) (missing documentation)
  • inversions(Permutation) -- see inversions -- computes the inversions of a permutation
  • isCartwrightSturmfels(Permutation) -- see isCartwrightSturmfels -- whether a permutation is Cartwright-Sturmfels
  • isCDG(Permutation) -- see isCDG -- whether a permutation is CDG.
  • isDerangement(Permutation) -- see isDerangement -- whether a permutation is a derangement
  • isEven(Permutation) -- see isEven -- whether a permutation is even
  • isOdd(Permutation) -- see isOdd -- whether a permutation is odd
  • isSeparable(Permutation) -- see isSeparable -- whether a permutation is separable.
  • isVexillary(Permutation) -- see isVexillary -- whether a permutation is vexillary
  • isWellDefined(Permutation) -- checks if a list is a valid permutation
  • length(Permutation) -- computes the length of a permutation
  • Matrix * Permutation -- computes the action of a permutation on the columns of a matrix
  • Matrix ^ Permutation -- computes the action of a permutation on a matrix
  • Matrix _ Permutation -- computes the action of a permutation on the rows of a matrix
  • matrix(Permutation) -- computes the matrix representation of a permutation
  • new Permutation from VisibleList (missing documentation)
  • ord(Permutation) -- see ord -- computes the order of a permutation
  • Permutation * Matrix -- computes the action of a permutation on the rows of a matrix
  • Permutation * VisibleList -- computes the action of a permutation on a list
  • Permutation == Permutation -- whether two permutations are the same
  • Permutation _ List -- selects a subset of the permutation when regarded as a list
  • Permutation _ Sequence -- selects a subset of the permutation when regarded as a list
  • Permutation _ ZZ -- selects an element from the permutation when regarded as a list
  • records(Permutation) -- see records -- computes the saliances of a permutation
  • reducedWords(Permutation) -- see reducedWords -- computes the reduced words of a permutation
  • saliances(Permutation) -- see saliances -- computes the saliances of a permutation
  • sign(Permutation) -- computes the sign of a permutation
  • strongBruhatOrder(Permutation,Permutation) -- see strongBruhatOrder -- compares two permutations in the (strong) Bruhat order
  • tex(Permutation) (missing documentation)
  • toString(Permutation) (missing documentation)
  • VisibleList _ Permutation -- computes the action of a permutation on a list
  • weakBruhatOrder(Permutation,Permutation) -- see weakBruhatOrder -- compares two permutations in the weak Bruhat order

For the programmer

The object Permutation is a type, with ancestor classes VisibleList < BasicList < Thing.


The source of this document is in Permutations/Documentation/mainDocs.m2:21:0.