Q-ROOK POLYNOMIALS AND MATRICES OVER FINITE-FIELDS

Connections between q-rook polynomials and matrices over finite fields are exploited to derive a new statistic for Garsia and Remmel's q-hit polynomial. Both this new statistic mat and another statistic for the q-hit polynomial xi recently introduced by Dworkin are shown to induc e different multiset Mahonian permutation statistics for any Ferrers b oard. In addition, for the triangular boards they are shown to generat e different families of Euler-Mahonian statistics. For these boards th e xi family includes Denert's statistic den, and gives a new proof of Foata and Zeilberger's Theorem that (exc, den) is equidistributed with (des, maj). The mat family appears to be new. A proof is also given t hat the q-hit polynomials are symmetric and unimodal. (C) 1998 Academi c Press.