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We consider the problem of finding the number of matrices over a finite fieldwith a certain rank and with support that avoids a subset of the entries. Thesematrices are a q-analogue of permutations with restricted positions (i.e., rookplacements). For general sets of entries these numbers of matrices are notpolynomials in q (Stembridge 98); however, when the set of entries is a Youngdiagram, the numbers, up to a power of q-1, are polynomials with nonnegativecoefficients (Haglund 98).  In this paper, we give a number of conditions under which these numbers arepolynomials in q, or even polynomials with nonnegative integer coefficients. Weextend Haglund's result to complements of skew Young diagrams, and we applythis result to the case when the set of entries is the Rothe diagram of apermutation. In particular, we give a necessary and sufficient condition on thepermutation for its Rothe diagram to be the complement of a skew Young diagramup to rearrangement of rows and columns. We end by giving conjecturesconnecting invertible matrices whose support avoids a Rothe diagram andPoincar\'e polynomials of the strong Bruhat order.

Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams