Unfair distributions counted by the generalized Stirling Numbers

Abstract

We investigate the generalized Stirling numbers S(n, k; α, β, γ) introduced by Hsu and Shiue from a combinatorial point of view. We present a combinatorial interpretation in terms of certain restricted distributions of labeled balls into unlabeled cells and a special cell where all cells are divided into distinct compartments. Using our interpretation, we find combinatorial proofs of several identities involving S(n, k; α, β, γ) and the associated generalized Bell numbers. Connections are made with some prior combinatorial models for the r-Lah numbers and other arrays, one via a sign-changing involution and another through a direct bijection. Finally, an additional parameter is introduced into our model which allows for further generalization.