Rook Theoretic Proofs of Some Identities Related to Spivey’s Bell Number Formula

Abstract

We use rook placements to prove Spivey’s Bell number formula and other identities related to it, in particular, some convolution identities involving Stirling numbers and relations involving Bell numbers. To cover as many special cases as possible, we work on the generalized Stirling numbers that arise from the rook model of Goldman and Haglund. An alternative combinatorial interpretation for the Type II generalized $q$-Stirling numbers of Remmel and Wachs is also introduced, in which the method used to obtain the earlier identities can be adapted easily.