Rook theory, generalized {S}tirling numbers and {\((p,q)\)}-analogues

J. B. Remmel ; Michelle L. Wachs

The electronic journal of combinatorics, Tome 11 (2004) no. 1

Résumé

In this paper, we define two natural $(p,q)$-analogues of the generalized Stirling numbers of the first and second kind $S^1(\alpha,\beta,r)$ and $S^2(\alpha,\beta,r)$ as introduced by Hsu and Shiue [Adv. in Appl. Math. 20 (1998), 366–384]. We show that in the case where $\beta =0$ and $\alpha$ and $r$ are nonnegative integers both of our $(p,q)$-analogues have natural interpretations in terms of rook theory and derive a number of generating functions for them. We also show how our $(p,q)$-analogues of the generalized Stirling numbers of the second kind can be interpreted in terms of colored set partitions and colored restricted growth functions. Finally we show that our $(p,q)$-analogues of the generalized Stirling numbers of the first kind can be interpreted in terms of colored permutations and how they can be related to generating functions of permutations and signed permutations according to certain natural statistics.

Détail

DOI : 10.37236/1837

Classification : 05A30, 05A05, 05A18
Mots-clés : generalized Stirling numbers, \((p,q)\)-analogues, rook placements, permutation statistics, colored partitions, restricted growth functions