Geodesic
A \((p, q)\)-analogue of the \(r\)-Whitney-Lah numbers
Journal of integer sequences, Tome 19 (2016) no. 5
In this paper, we consider a $(p, q)$-generalization of the $r$-Whitney-Lah numbers that reduces to these recently introduced numbers when $p = q = 1$. We develop a combinatorial interpretation for our generalized numbers in terms of a pair of statistics on an extension of the set of $r$-Lah distributions wherein certain elements are assigned a color. We obtain generalizations of some earlier results for the $r$-Whitney-Lah sequence, including explicit formulas and various recurrences, as well as ascertain some new results for this sequence. We provide combinatorial proofs of some additional formulas in the case when $q = 1$, among them one that generalizes an identity expressing the $r$-Whitney-Lah numbers in terms of the $r$-Lah numbers. Finally, we introduce the $(p, q)$-Whitney-Lah matrix and study some of its properties.
Classification : 05A19, 05A15, 15A23
Keywords: r-Whitney-lah number, r-lah number, q-generalization, combinatorial proof 
@article{JIS_2016__19_5_a2,
     author = {Ram{\'\i}rez,  Jos\'e L. and Shattuck,  Mark},
     title = {A \((p, q)\)-analogue of the {\(r\)-Whitney-Lah} numbers},
     journal = {Journal of integer sequences},
     year = {2016},
     volume = {19},
     number = {5},
     zbl = {1342.05015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2016__19_5_a2/}
}
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Ramírez,  José L.; Shattuck,  Mark. A \((p, q)\)-analogue of the \(r\)-Whitney-Lah numbers. Journal of integer sequences, Tome 19 (2016) no. 5. http://geodesic.mathdoc.fr/item/JIS_2016__19_5_a2/