A combinatorial interpretation of the p,qp,q-hit numbers

Briggs and Remmel [K.S. Briggs, J.B. Remmel, A p,qp,q-analogue of a formula of Frobenius, Electron. J. Combin. 10 (1) (2003) #R9] defined a p,qp,q-analogue of the hit numbers and showed that they are polynomials in pp and qq with nonnegative coefficients for all Ferrers boards. Here we show that there is a natural extension of Dworkin’s statistic ξξ [M. Dworkin, An interpretation for Garsia and Remmel’s qq-hit numbers, J. Combin. Theory Ser. A 81 (1998) 149–175] so that for Ferrers boards, the p,qp,q-hit numbers introduced by Briggs and Remmel arise by p,qp,q-counting placements of nn nonattacking rooks in the n×nn×n board. Our proofs are based on different methods than those used by either Dworkin or Haglund [J. Haglund, qq-Rook polynomials and matrices over finite fields, Adv. in Appl. Math. 20 (1998) 450–487]. Our proofs are based on showing that our combinatorial interpretations of the p,qp,q-hit numbers satisfy certain recursions introduced by Briggs and Remmel.