Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.6

A (p, q)-Analogue of the r-Whitney-Lah Numbers

José L. Ramírez
Departamento de Matemáticas
Universidad Sergio Arboleda
110221 Bogotá

Mark Shattuck
Department of Mathematics
University of Tennessee
Knoxville, TN 37996


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.

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(Concerned with sequences A008297 A143497.)

Received January 16 2016; revised version received May 24 2016. Published in Journal of Integer Sequences, June 3 2016.

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