A p, q-analogue of a Formula of Frobenius

Garsia and Remmel (JCT. A 41 (1986), 246-275) used rook configurations 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 defined generalized p, q-Stirling numbers of the first and second kind in terms of rook placements. Additionally, they extended their definition to give a p, q-analogue of rook numbers for arbitrary Ferrers boards. In this paper, we use Remmel and Wach’s definition 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 Sn. We further define 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 coefficients.