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Permutations -- a package which implements permutations

Description

This package defines the Permutation type. An overview of the package's can be found in the Permutations Guide.

Author

Version

This documentation describes version 1.1 of Permutations.

Citation

If you have used this package in your research, please cite it as follows:

@misc{PermutationsSource,
  title = {{Permutations: functions for working with permutations. Version~1.1}},
  author = {Sean Grate},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

  • Types
  • Functions and commands
    • ascendingRuns -- computes the ascending runs of a permutation
    • ascents -- computes the ascents of a permutation
    • avoidsPattern -- whether a permutation avoids a pattern
    • avoidsPatterns -- whether a permutation simultaneously avoids a list of patterns
    • cycleDecomposition -- computes the decomposition of a permutation as a product of disjoint cycles
    • cycleType -- computes the cycle type of a permutation
    • descendingRuns -- computes the descending runs of a permutation
    • descents -- computes the descents of a permutation
    • exceedances -- computes the exceedances of a permutation
    • fixedPoints -- computes the fixed points of a permutation
    • foataBijection -- computes the image of a permutation under the Foata bijection
    • inversions -- computes the inversions of a permutation
    • isCartwrightSturmfels -- whether a permutation is Cartwright-Sturmfels
    • isCDG -- whether a permutation is CDG.
    • isDerangement -- whether a permutation is a derangement
    • isEven -- whether a permutation is even
    • isOdd -- whether a permutation is odd
    • isSeparable -- whether a permutation is separable.
    • isVexillary -- whether a permutation is vexillary
    • ord -- computes the order of a permutation
    • permutation -- see Permutation -- the Permutation type
    • randomPermutation -- generates a random permutation
    • records -- computes the saliances of a permutation
    • reducedWords -- computes the reduced words of a permutation
    • saliances -- computes the saliances of a permutation
    • strongBruhatOrder -- compares two permutations in the (strong) Bruhat order
    • symmetricGroupPoset -- constructs the poset of the symmetric group under a partial order
    • transposition -- constructs a transposition
    • weakBruhatOrder -- compares two permutations in the weak Bruhat order
  • Methods
    • 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,ZZ) -- see extend -- rewrites a permutation as a permutation on more symbols
    • 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
    • foataBijection(Permutation) -- see foataBijection -- computes the image of a permutation under the Foata bijection
    • html(Permutation) (missing documentation)
    • inverse(Permutation) -- computes the inverse of a permutation
    • 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(VisibleList) -- see Permutation -- the Permutation type
    • Permutation * Matrix -- computes the action of a permutation on the rows of a matrix
    • Permutation * Permutation -- computes the product of two permutations
    • Permutation * VisibleList -- computes the action of a permutation on a list
    • Permutation == Permutation -- whether two permutations are the same
    • Permutation ^ ZZ -- computes the power of a permutation
    • 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
    • randomPermutation(ZZ) -- see randomPermutation -- generates a random permutation
    • 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
    • symmetricGroupPoset(ZZ,Function) -- see symmetricGroupPoset -- constructs the poset of the symmetric group under a partial order
    • tex(Permutation) (missing documentation)
    • toString(Permutation) (missing documentation)
    • 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
    • VisibleList _ Permutation -- computes the action of a permutation on a list
    • weakBruhatOrder(Permutation,Permutation) -- see weakBruhatOrder -- compares two permutations in the weak Bruhat order
  • Symbols
    • Weak (missing documentation)
    • Side -- see weakBruhatOrder -- compares two permutations in the weak Bruhat order

For the programmer

The object Permutations is a package, defined in Permutations.m2, with auxiliary files in Permutations/.

The source of this document is in Permutations/Documentation/packageDocs.m2:12:0.