Generalizations of Bell number formulas of Spivey and Mezo

TL;DR

This paper introduces q-generalizations of Spivey's Bell number formula, deriving new identities involving q-Stirling and q-Lah numbers, and extends recent r-Stirling formulas with combinatorial proofs and interpretations.

Contribution

It provides the first q-analogues of Spivey's Bell number formula in various combinatorial settings, generalizing existing formulas and offering new combinatorial interpretations.

Findings

Derived q-analogues of Spivey's Bell number formula.

Established new identities involving q-Stirling and q-Lah numbers.

Provided combinatorial proofs and interpretations for generalized Stirling numbers.

Abstract

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 Mezo. 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.