data

This page contains data supporting Conjecture 5.1 in [2] which (reformulated) states the following:

Conjecture (Conj. 5.1 in [2]): For a permutation w=w1w2....wn in Sn, let Dw be the set of co-inversions Dw = { (i,j) | 1<= i < j <= n, wi < wj }. Then the number matq(n,Dw,r) of n-by-n rank r matrices A over Fq with forced zeros Aij if (i,j) is in Dw is of the form g(q)(q-1)r where g(q) is a polynomial with nonnegative integer coefficients.

This conjecture has been verified for:

• • 0 <= r <= n, and n<= 6, and for the invertible case r=n for n=7,8,9. (as of July 2013)

  • all co-vexillary permutations w (w avoiding 3142) due to a result by Haglund [1].

  • all skew co-vexillary permutations w (w avoiding 9 patterns in S5 and S6) due to Theorem 5.4 in [2].

  • all Gasharov-Reiner permutations w (w avoiding 4231, 35142, 42531, and 351624) due to Theorem 3.1 in [3].

UPDATE (July 2017): The polynomiality part of the conjecture is true but there is an example in n=10 that gives a polynomial g(q) with not all coefficients nonnegative [4] 

References:

• 1. J. Haglund, q-rook polynomials and matrices over finite fields, Adv. in Appl. Math., 20(4):450-487, 1998.

  2. A. Klein, J. B. Lewis, A.H. Morales, Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams,reJ. Algebraic Combinatorics 39 (2014), pp. 429-456, arXiv:1203.5804

  3. J.B. Lewis, A.H. Morales, Combinatorics of diagrams of permutations, 2014,  arXiv:1405.1608

  4. J.B. Lewis, A.H. Morales, Rook theory for the finite general linear group, 2017, arxiv:1707.08192

The files on this page include tables of lists. The format is lists of [w, f(q)] where f(q) is a polynomial in q: 

Tables of Poincaré polynomials Pw(q)=sum_(u>=w) qlen(u), for permutations u in the Strong Bruhat order.

• every permutation w is a permutation of length n for n = 3, 4, 5, 6. These are the files "allpoinS3.txt", "allpoinS4.txt", "allpoinS5.txt", "allpoinS6.txt" . 

• permutations in S7, modulo inverse permutations and reverse complements, that contain any of the patterns 1324, 24153, 31524, and 426153.

Tables of matq(n,Rw,r)/(q – 1)r , where matq(n,Rw,r) is the number of rank r matrices whose support avoids the Rothe diagram of w.

• every permutation w is a permutation of length n for n = 3 to n=6. These are the files "allRotheS3.txt", "allRotheS4.txt", "allRotheRankS5.txt", "allRotheRankS6.txt".

• permutations w of length n = 7, modulo inverse and reverse complements. This is the file "allRotheRankS7.txt"

• permutations w of length n = 8 (only for r=8), those modulo inverses and reverse complements, that contain any of the patterns 1324, 24153, 31524, and 426153). This is the file "allRotheInvS8.txt".

• permutations w of length n = 9 (only for r=9), those modulo inverses and reverse complements, that contain any of the patterns 1324, 24153, 31524, and 426153). This is the file "allRotheInvS9.tgz" (compressed file, to uncompress use command > tar -zxvf allRotheInvS9.tgz ).

Tables of q-rook numbers Rn(n,LHw,,SE)= sum_(C) qinv(C,S,SE) where the sum is over placements of n-rooks on S and inv(C,S,SE) is the number of SE-inversions of C in S.

• every permutation w is a permutation of length n for n = 3, 4, 5, 6. These are the files "rknumS3.txt", "rknumS4.txt", "rknumS5.txt", "rknumS6.txt".

The data can be read in maple with the command  > read "filename.txt": 

Acknowledgements: We thank Franco Saliola for computation time on his FRQNT funded machine where part of this data came from.