A \(p,q\)-analogue of a formula of Frobenius
type: Item
instance of: scholarly article
title: A \(p,q\)-analogue of a formula of Frobenius
zbMATH Open document ID: 1011.05010
author: Karen S. Briggs
author: Jeffrey Remmel
published in: The Electronic Journal of Combinatorics
publication date: 2003-05-07T00:00:00Z
full work available at URL: https://eudml.org/doc/122599
full work available at URL: v10i1r9.html
review text: 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.
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 05E15
Mathematics Subject Classification ID: 05A18
zbMATH DE Number: 1907094
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1907094