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To count matrices over finite fields by rank and with restricted support we use procedures built around Stembridge's reduce package. You will need the files "reduce.txt" and "kc_test.txt". 

The text file "matqresuppV1-2.txt" has the following main procedures:

• matq(m,n,S,r) computes the number of m-by-n matrices of rank r matrices whose support avoids the complement of S.  In particular, the function matq(m × n, S, r) from paper would be  matq(m,n,compsh(m,n,S),r).  REQUIREMENTS "reduce.txt" and "kc_test.txt" from Stembridge's reduce package. Must input/read these files.

• qRkNum(m,n,r,S,q,dir) computes the q-rook number sum_C q^(inv^(dir)(C,S))  where inv^(dir)(C,S) is the number of dir-inversions on S and dir is NE or NW or SE or SW and C is a rook placement on S of r rooks.

• fastMatqRothe(w) computes matq(n,n,Rw,n) where Rw is the Rothe diagram of w. This procedure removes rows and columns full of free entries to speed up computations REQUIREMENTS "reduce.txt" and "kc_test.txt" from Stembridge's reduce package. Must read/input these files.

• getRP(n,r,S) computes the set of r-partial permutations that avoid subset S of [n] × [n].

• strsh(m,n,lam) returns the subset of [m] × [n] indexed by partition lam, option 'nocomp' at the end gives  complement.

• skewsh(m,n,lam,mu) returns the subset of [m] × [n] indexed by skew partition lambda/mu, option 'nocomp' at the end gives the complement of the skew shape.

• Rothesh(w) returns the SE-Rothe diagram of the permutation w on n letters.

• LHullsh(w) returns the left hull of the permutation w.

• compsh(m,n,S) returns complement of set S in [m] × [n].

• getMatrixXs(m,n,S) returns [X,nument] where X is the m × n matrix (x[i,j]) such that x[i,j]=0 if [i,j] is in S and nument is the number of free entries in X, i.e., n*m - #S.

NEW procedure in version 1.2:

• matqBruhat(n,S,r) computes the number of n-by-n rank r matrices whose support avoids S using intersection with Bruhat cells of permutations (in the case r=n) or partial permutations (in the case r<n). This procedure appears to be faster than matq(n,n,S,r). This algorithm was suggested by Alexander Woo and it was used to compute the number of rank r 7-by-7 matrices whose support avoids the incidence matrix of the Fano plane.

new procedure in version 1.1: 

• recurmatq(m,n,S,r) computes RECURSIVELY the number of m-by-n rank r matrices whose supports avoids COMPLEMENT OF S. REQUIREMENT: set S in [m] by [n] needs to satisfy the following property: either S or its transpose contains at most two entries in a row OR the  complement(S) or its transpose contains at most two entries in a row (uses Theorems 3.5 and 3.7 from here).

There is also a maple worksheet "suppMatResPos.mw.xml" with these functions and most of the examples and figures from the paper.