Coefficients of Gaussian Polynomials Modulo $N$

Abstract

Let $\left[{n \atop k}\right]_q$ be a $q$-binomial coefficient. Stanley conjectured that the function $f_{k,R}(n) = \left|\left\{\alpha : [q^{\alpha}] \left[{n \atop k}\right]_q  \equiv R \pmod{N}\right\}\right|$ is quasi-polynomial for $N$ prime. We prove this for any integer $N$ and obtain an expression for the generating function $F_{k,R}(x)$ for $f_{k,R}(n)$.