Joel Brewster Lewis

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

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 in Ann. Comb. 2(4):365, 1998); 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 in Adv. Appl. Math. 20(4):450, 1998).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.

Matrices with restricted entries and

We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are

q

q

-analogues of permutations

-analogues of permutations with certain restricted values. We obtain a simple closed formula for the number of invertible matrices with zero diagonal, a

Enumeration of graded (3+1)-avoiding posets

q

Absolute order in general linear groups

-analogue of derangements, and a curious relationship between invertible skew-symmetric matrices and invertible symmetric matrices with zero diagonal. In addition, we provide recursions to enumerate matrices and symmetric matrices with zero diagonal by rank, and we frame some of our results in the context of Lie theory. Finally, we provide a brief exposition of polynomiality results for enumeration questions related to those mentioned, and give several open questions.

Invariants of GLn(ð”½q) in polynomials modulo Frobenius powers

Abstract The notion of ( 3 + 1 ) -avoidance has shown up in many places in enumerative combinatorics, but the natural goal of enumerating all ( 3 + 1 ) -avoiding posets remains open. In this paper, we enumerate graded ( 3 + 1 ) -avoiding posets for both reasonable definitions of the word “graded.” Our proof consists of a number of structural theorems followed by some generating function computations. We also provide asymptotics for the growth rate of the number of graded ( 3 + 1 ) -avoiding posets.

GL n ( F q ) -analogues of factorization problems in the symmetric group

This paper studies a partial order on the general linear group GL(V) called the absolute order, derived from viewing GL(V) as a group generated by reflections, that is, elements whose fixed space has codimension one. The absolute order on GL(V) is shown to have two equivalent descriptions, one via additivity of length for factorizations into reflections, the other via additivity of fixed space codimensions. Other general properties of the order are derived, including self-duality of its intervals. Working over a finite field F_q, it is shown via a complex character computation that the poset interval from the identity to a Singer cycle (or any regular elliptic element) in GL_n(F_q) has a strikingly simple formula for the number of chains passing through a prescribed set of ranks.

Combinatorial Aspects of Flashcard Games

Conjectures are given for Hilbert series related to polynomial invariants of finite general linear groups, one for invariants mod Frobenius powers of the irrelevant ideal, one for cofixed spaces of polynomials.

Baron Münchhausen Redeems Himself: Bounds for a Coin-Weighing Puzzle

We consider GL n ( F q ) -analogues of certain factorization problems in the symmetric group S n : rather than counting factorizations of the long cycle ( 1 , 2 , ź , n ) given the number of cycles of each factor, we count factorizations of a regular elliptic element given the fixed space dimension of each factor. We show that, as in S n , the generating function counting these factorizations has attractive coefficients after an appropriate change of basis. Our work generalizes several recent results on factorizations in GL n ( F q ) and also uses a character-based approach.As an application of our results, we compute the asymptotic growth rate of the number of factorizations of fixed genus of a regular elliptic element in GL n ( F q ) into two factors as n ź ∞ . We end with a number of open questions.