$q$-Stirling numbers revisited

TL;DR

This paper provides combinatorial proofs of various $q$-Stirling identities using restricted growth words, introduces a two-parameter generalization, and unifies multiple existing identities with new proofs and perspectives.

Contribution

It offers new combinatorial proofs, a unifying two-parameter generalization, and a symmetric function version of $q$-Stirling identities, enhancing understanding and connections among these identities.

Findings

Combinatorial proofs of $q$-Stirling identities using restricted growth words.

A two-parameter generalization unifying identities of Mercier.

A symmetric function version of the $q$-Stirling identities.

Abstract

We give combinatorial proofs of $q$-Stirling identities using restricted growth words. This includes a poset theoretic proof of Carlitz's identity, a new proof of the $q$-Frobenius identity of Garsia and Remmel and of Ehrenborg's Hankel $q$-Stirling determinantal identity. We also develop a two parameter generalization to unify identities of Mercier and include a symmetric function version.