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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)      .

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