A p,q-analogue of a Formula of Frobenius

Garsia and Remmel (JCT. A 41 (1986), 246-275) used rook con gurations to give a combinatorial interpretation to the q-analogue of a formula of Frobenius relating the Stirling numbers of the second kind to the Eulerian polynomials. Later, Remmel and Wachs de ned generalized p; q-Stirling numbers of the rst and second kind in terms of rook placements. Additionally, they extended their de nition to give a p; q-analogue of rook numbers for arbitrary Ferrers boards. In this paper, we use Remmel and Wach's de nition and an extension of Garsia and Remmel's proof to give a combinatorial interpretation to a p; q-analogue of a formula of Frobenius relating the p; q-Stirling numbers of the second kind to the trivariate distribution of the descent number, major index, and comajor index over S n . We further de ne a p; q-analogue of the hit numbers, and show analytically that for Ferrers boards, the p; q-hit numbers are polynomials in (p; q) with nonnegative coecients