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

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José L. Ramírez

Departamento de Matemáticas

Universidad Sergio Arboleda

110221 Bogotá

Colombia

Mark Shattuck

Department of Mathematics

University of Tennessee

Knoxville, TN 37996

USA

**Abstract:**

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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