Generalized Binomial Coefficients and the Subset–Subspace Problem

Abstract

Generalized binomial coefficients of the first and second kind are defined in terms of object selection with and without repetition from weighted boxes. The combinatorial definition unifies the binomial coefficients, the Gaussian coefficients, and the Stirling numbers and their recurrence relations under a common interpretation. Combinatorial proofs for some Gaussian coefficient identities are derived and shown to reduce to the ordinary binomial coefficients whenq=1. This approach provides a different perspective on the subset–subspace analogy problem. Generating function relations for the generalized binomial coefficients are derived by formal methods.