Geodesic
Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams
Journal of Algebraic Combinatorics, Tome 39 (2014) no. 2, pp. 429-456.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

We consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a $q$-analogue of permutations with restricted positions (i.e., rook placements). For general sets of entries, these numbers of matrices are not polynomials in $q$ [J. R. Stembridge, Ann. Comb. 2, No. 4, 365--385 (1998; Zbl 0927.05002)]; however, when the set of entries is a Young diagram, the numbers, up to a power of $q-1$, are polynomials with nonnegative coefficients [J. Haglund, Adv. Appl. Math. 20, No. 4, 450--487 (1998; Zbl 0914.05002)]. In this paper, we give a number of conditions under which these numbers are polynomials in $q$, or even polynomials with nonnegative integer coefficients. We extend Haglund's result to complements of skew Young diagrams, and we apply this result to the case where the set of entries is the Rothe diagram of a permutation. In particular, we give a necessary and sufficient condition on the permutation for its Rothe diagram to be the complement of a skew Young diagram up to rearrangement of rows and columns. We end by giving conjectures connecting invertible matrices whose support avoids a Rothe diagram and Poincaré polynomials of the strong Bruhat order.
Classification : 05A15, 05E10
Keywords: rook placements, finite fields, Bruhat order, Rothe diagrams, pattern avoidance, $q$-analogues 
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     author = {Klein, Aaron J. and Lewis, Joel Brewster and Morales, Alejandro H.},
     title = {Counting matrices over finite fields with support on skew {Young} diagrams and complements of {Rothe} diagrams},
     journal = {Journal of Algebraic Combinatorics},
     pages = {429--456},
     publisher = {mathdoc},
     volume = {39},
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     year = {2014},
     language = {en},
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Klein, Aaron J.; Lewis, Joel Brewster; Morales, Alejandro H. Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams. Journal of Algebraic Combinatorics, Tome 39 (2014) no. 2, pp. 429-456. http://geodesic.mathdoc.fr/item/JAC_2014__39_2_a2/