Journal of Algebraic Combinatorics

, Volume 39, Issue 2, pp 429–456

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

  • Aaron J. Klein
  • Joel Brewster Lewis
  • Alejandro H. Morales
Article

DOI: 10.1007/s10801-013-0453-x

Cite this article as:
Klein, A.J., Lewis, J.B. & Morales, A.H. J Algebr Comb (2014) 39: 429. doi:10.1007/s10801-013-0453-x
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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 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.

Keywords

Rook placementsFinite fieldsBruhat orderRothe diagramsPattern avoidanceq-analogues

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Aaron J. Klein
    • 1
  • Joel Brewster Lewis
    • 2
  • Alejandro H. Morales
    • 3
  1. 1.Brookline High SchoolBrooklineUSA
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  3. 3.Laboratoire de Combinatoire et d‘Informatique Mathématique (LaCIM)Université du Québec à MontréalMontréalCanada