Coefficients of Gaussian Polynomials Modulo $N$

Abstract

The $q$-analogue of the binomial coefficient, known as a $q$-binomial coefficient, is typically denoted ${\left[\genfrac{}{}{0ex}{}{n}{k}\right]}_q$. These polynomials are important combinatorial objects, often appearing in generating functions related to permutations and in representation theory. Stanley conjectured that the function $f_{k,R}(n)=\#\left\{i:[q^i]{\left[\genfrac{}{}{0ex}{}{n}{k}\right]}_q\equiv R(\mathrm{mod}N)\right\}$ is quasipolynomial for $N=2$. We generalize, showing that this is in fact true for any integer $N\in \mathbb{N}$ and determine a quasi-period ${\pi }_N^{\prime }(k)$ derived from the minimal period ${\pi }_N(k)$ of partitions with at most $k$ parts modulo $N$.