Journal of Combinatorics

Volume 2 (2011)

Number 3

Matrices with restricted entries and $q$-analogues of permutations

Pages: 355 – 395

DOI: http://dx.doi.org/10.4310/JOC.2011.v2.n3.a2

Authors

Joel Brewster Lewis (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Ricky Ini Liu (University of Michigan, Ann Arbor, Mich., U.S.A.)

Alejandro H. Morales (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Greta Panova (Department of Mathematics, University of California at Los Angeles)

Steven V. Sam (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Yan X. Zhang (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Abstract

We study the functions that count matrices of given rank over a finitefield with specified positions equal to zero. We show that thesematrices are $q$-analogues of permutations with certain restrictedvalues. We obtain a simple closed formula for the number of invertiblematrices with zero diagonal, a $q$-analogue of derangements, anda curious relationship between invertible skew-symmetric matricesand invertible symmetric matrices with zero diagonal. In addition,we provide recursions to enumerate matrices and symmetric matriceswith zero diagonal by rank, and we frame some of our resultsin the context of Lie theory. Finally, we provide a brief expositionof polynomiality results for enumeration questions related to thosementioned, and give several open questions.

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