$q$-Stirling numbers revisited

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.