A 𝑞-analog of restricted growth functions, Dobinski’s equality, and Charlier polynomials

We apply finite operator techniques due to G. C. Rota to a combinatorial identity, which counts a collection of generalized restricted growth functions in two ways, and obtain a q -analog of Charlier polynomials and Dobinski’s equality for the number of partitions of an n -set. Our methods afford a unified proof of certain identities in the combinatorics of finite dimensional vector spaces over GF ( q ) {\text {GF}}(q) .