{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:18:40Z","timestamp":1759335520887,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.37236\/1702","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T21:29:46Z","timestamp":1578691786000},"source":"Crossref","is-referenced-by-count":8,"title":["A $p,q$-analogue of a Formula of Frobenius"],"prefix":"10.37236","volume":"10","author":[{"given":"Karen S.","family":"Briggs","sequence":"first","affiliation":[]},{"given":"Jeffrey B.","family":"Remmel","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2003,3,18]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v10i1r9\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v10i1r9\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T00:09:47Z","timestamp":1579306187000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v10i1r9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,3,18]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2003,1,6]]}},"URL":"https:\/\/doi.org\/10.37236\/1702","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2003,3,18]]},"article-number":"R9"}}