A \(p,q\)-analogue of a formula of Frobenius
Summary: \textit{A. M. Garsia} and \textit{J. B. Remmel} [J. Comb. Theory, Ser. A 41, 246-275 (1986; Zbl 0598.05007)] 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 \(S_n\). 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.
scientific article; zbMATH DE number 1907094