Generalizing the Combinatorics of Binomial Coefficients via !-nomials

An !-sequence is defined by an = !an−1 − an−2, with initial conditions a0 = 0, a1 = 1. These !-sequences play a remarkable role in partition theory, allowing !generalizations of the Lecture Hall Theorem and Euler’s Partition Theorem. These special properties are not shared with other sequences, such as the Fibonacci sequence, defined by second-order linear recurrences. The !-sequence gives rise to the !-nomial coefficient (n k )(!) = ∏k i=1(an+1−i/ai), which is known to be an integer.In this paper, we use algebraic and combinatorial properties of !-sequences to interpret the !-nomial coefficients in terms of weighted lattice paths, integer partitions, and probablility distributions. We show how to use these interpretations to uncover !-generalizations of familiar hypergeometric identities involving binomial coefficients. This leads naturally to an !-analogue of the q-binomial coefficients (Gaussian polynomials) and a corresponding generalization of the “partitions in a box” interpretation of ordinary q-binomial coefficients.