Geodesic
A \(p,q\)-analogue of a formula of Frobenius
The electronic journal of combinatorics, Tome 10 (2003)
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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 $S_n$. 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.
DOI : 10.37236/1702
Classification : 05A30, 05A05, 05A19, 05A18 
@article{10_37236_1702,
     author = {Karen S. Briggs and Jeffrey B. Remmel},
     title = {A \(p,q\)-analogue of a formula of {Frobenius}},
     journal = {The electronic journal of combinatorics},
     year = {2003},
     volume = {10},
     doi = {10.37236/1702},
     zbl = {1011.05010},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1702/}
}
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Karen S. Briggs; Jeffrey B. Remmel. A \(p,q\)-analogue of a formula of Frobenius. The electronic journal of combinatorics, Tome 10 (2003). doi: 10.37236/1702

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