Rook Theory, Generalized Stirling Numbers and $(p,q)$-Analogues

Rook Theory, Generalized Stirling Numbers and (p, q)-Analogues

In this paper, we define two natural (p, q)-analogues of the generalized Stirling numbers of the first and second kind S(α, β, r) and S(α, β, r) as introduced by Hsu and Shiue [17]. We show that in the case where β = 0 and α 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 al...

Stirling Numbers and Generalized Zagreb Indices

We show how generalized Zagreb indices $M_1^k(G)$ can be computed by using a simple graph polynomial and Stirling numbers of the second kind. In that way we explain and clarify the meaning of a triangle of numbers used to establish the same result in an earlier reference.

Generalized Stirling and Lah numbers

The theory of modular binomial lattices enables the simultaneous combinatorial analysis of finite sets, vector spaces, and chains. Within this theory three generalizations of Stifling numbers of the second kind, and of Lah numbers, are developed. 1. Stirling numbers and their formal generalizations The nota t ional convent ions of this paper are as follows: N = {0,1,2 . . . . }, P = {1,2,. . . ...

A Rook Theory Model for the Generalized p, q-Stirling Numbers of the First and Second Kind

In (EJC 11 (2004), #R84), Remmel and Wachs presented two natural ways to define p, qanalogues of the generalized Stirling numbers of the first and second kind, S(α, β, r) and S(α, β, r) as introduced by Hsu and Shiue (Adv. App. Math 20 (1998), 366-384). In this paper, we present a rook theoretic model for each type of p, q-analogue based on a pair of boards parametrized by the nonnegative integ...

Generalized higher order Stirling numbers