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# MatrixSchubert : Table of Contents

- MatrixSchubert -- matrix Schubert varieties and ASM varieties
  - Investigating matrix Schubert varieties -- basic functions for Schubert determinantal ideals
  - Investigating ASM varieties -- basic functions for alternating sign matrix ideals
    - padASM -- pad an ASM with an identity matrix
    - isPartialASM -- whether a matrix is a partial alternating sign matrix
    - partialASMToASM -- extend a partial alternating sign matrix to an alternating sign matrix
    - rankTable -- compute a table of rank conditions that determines the corresponding ASM or matrix Schubert variety
    - rotheDiagram -- find the Rothe diagram of a partial alternating sign matrix
    - augmentedRotheDiagram -- find the Rothe diagram and rank table for a partial ASM or permutation
    - essentialSet -- compute the essential set in the Rothe Diagram for a partial ASM or a permutation.
    - augmentedEssentialSet -- find the essential set and rank conditions for a partial ASM or a permutation
    - schubertDeterminantalIdeal -- compute an alternating sign matrix ideal (for example, a Schubert determinantal ideal)
    - fultonGens -- compute the Fulton generators of an ASM ideal (for example, a Schubert determinantal ideal)
    - entrywiseMinRankTable -- compute the entrywise minimum rank table of a list of ASMs
    - entrywiseMaxRankTable -- compute the entrywise maximum rank table of a list of ASMs
    - schubertDecompose -- finds the decomposition of an ASM ideal into Schubert determinantal ideals
    - permSetOfASM -- finds the permutation set of an alternating sign matrix
    - isASMIdeal -- whether an ideal is an ASM ideal
    - isASM -- whether a matrix is an ASM
    - isASMUnion -- whether the union of matrix Schubert varieties is an ASM variety
    - getASM -- get the ASM of an ideal (if it exists)
    - isMinRankTable -- whether a matrix is the canonical rank table of some partial ASM
    - rankTableToASM -- to find the a partial ASM associated to a given rank table
    - rankTableFromMatrix -- returns the minimal rank table from an arbitrary integer matrix
    - schubertIntersect -- compute the intersection of ASM ideals
    - schubertAdd -- compute the sum of ASM ideals
    - ASMToMonotoneTriangle -- converts an ASM to a monotone triangle
    - monotoneTriangleToASM -- converts a monotone triangle to an ASM
    - ASMFullList -- lists all ASMs of a fixed size
    - ASMRandomList -- lists random ASMs of a fixed size
    - cohenMacaulayASMsList -- lists all Cohen-Macaulay ASMs of a fixed size which are not permutation matrices
    - nonCohenMacaulayASMsList -- lists all non-Cohen-Macaulay ASMs of a fixed size
    - isIntersectionOfSchubertDeterminantalIdeals -- whether an ideal is the intersection of Schubert determinantal ideals
    - isSchubertCM -- whether an ASM variety is Cohen-Macaulay
    - KPolynomialASM -- compute the K Polynomial of an ASM variety
    - schubertCodim -- compute the codimension (i.e., height) of a Schubert determinantal ideal or ASM ideal
    - schubertRegularity -- compute the Castelnuovo-Mumford regularity of the quotient by an ASM ideal
  - Initial ideals of ASM ideals -- basic functions for investigating initial ideals of ASM varieties
    - antiDiagInit -- compute the (unique) antidiagonal initial ideal of an ASM ideal
    - diagLexInitSE -- Diagonal initial ideal of an ASM ideal with respect to lex, starting from SE corner
    - diagLexInitNW -- Diagonal initial ideal of an ASM ideal with respect to lex, starting from NW corner
    - diagRevLexInit -- Diagonal initial ideal of an ASM ideal with respect to revlex, ordering variables from NW corner
    - subwordComplex -- to find the subword complex associated to w
    - initialIdealsList -- lists all antidiagonal initial ideals of ASMs of a fixed size
  - Functions for investigating permutations -- basic functions for permutations
    - isPerm -- whether a list is a permutation in 1-line notation
    - permToMatrix -- converts a permutation in 1-line notation into a permutation matrix
    - lastDescent -- finds the location of the last descent of a permutation
    - firstDescent -- finds the location of the first descent of a permutation
    - descentSet -- the descent set of a permutation
    - permLength -- to find the length of a permutation in 1-line notation.
    - inverseOf -- to return the inverse of a permutation in 1-line notation.
    - longestPerm -- to return the longest permutation of length n
    - toOneLineNotation -- rewrites a transposition or permutation matrix in 1-line notation
    - composePerms -- computes the composition of two permutations
    - isPatternAvoiding -- whether a permutation avoids certain patterns, e.g. $2143$-avoiding or $312$- and $231$-avoiding
    - isVexillary -- whether a permutation is vexillary, i.e. 2143-avoiding
    - avoidsAllPatterns -- whether a permutation avoids all of the given patterns
    - isCartwrightSturmfels -- whether a permutation is Cartwright-Sturmfels
    - isCDG -- whether a permutation is CDG
    - rajcode -- finds the Rajchgot code of a permutation
    - rajIndex -- finds the Rajchgot index of a permutation
    - grothendieckPolynomial -- computes the Grothendieck polynomial of a permutation
    - schubertPolynomial -- computes the Schubert polynomial of a permutation
    - doubleSchubertPolynomial -- computes the double Schubert polynomial of a permutation
    - PipeDream -- the class representing a pipe dream
      - pipeDreams -- computes the set of reduced pipe dreams corresponding to a permutation
      - pipeDreamsNonReduced -- computes the set of all pipe dreams corresponding to a permutation