Rook theoretic proofs of some identities related to Spivey's Bell number formula

TL;DR

This paper employs rook placements to prove Spivey's Bell number formula and related identities, extending to generalized Stirling numbers and providing new combinatorial interpretations.

Contribution

It introduces rook-theoretic proofs for Bell number identities, including generalized Stirling numbers and a new combinatorial interpretation for Type II q-Stirling numbers.

Findings

Proved Spivey's Bell number formula using rook placements

Extended identities to generalized Stirling numbers from rook models

Provided a new combinatorial interpretation for Type II q-Stirling numbers

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.