Generalizations of Bell Number Formulas of Spivey and Mező

Mark Shattuck

We provide $q$-generalizations of Spivey's Bell number formula in various settings by considering statistics on different combinatorial structures. This leads to new identities involving $q$-Stirling numbers of both kinds and $q$-Lah numbers. As corollaries, we obtain identities for both binomial and $q$-binomial coefficients. Our results at the same time also generalize recent $r$-Stirling number formulas of Mező. Finally, we provide a combinatorial proof and refinement of Xu's extension of Spivey's formula to the generalized Stirling numbers of Hsu and Shiue. To do so, we develop a combinatorial interpretation for these numbers in terms of extended Lah distributions.