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

TL;DR

This paper studies the enumeration of finite field matrices with support restrictions related to Young diagrams and Rothe diagrams, extending known polynomiality results and exploring connections to permutation properties and algebraic structures.

Contribution

It extends polynomiality results for matrix counts to complements of skew Young diagrams and Rothe diagrams, providing new conditions and conjectures linking these counts to permutation properties.

Findings

Numbers are polynomials in q for certain diagram classes

Extended Haglund's polynomiality result to complements of skew Young diagrams

Connected matrix support conditions to permutation properties and Bruhat order

Abstract

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 (Stembridge 98); however, when the set of entries is a Young diagram, the numbers, up to a power of q-1, are polynomials with nonnegative coefficients (Haglund 98). 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 when 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 Poincar\'e polynomials of the strong Bruhat order.