Coefficients of Gaussian Polynomials Modulo \(N\)

The \(q\)-analogue of the binomial coefficient, known as a \(q\)-binomial coefficient, is typically denoted \(\left[{n \atop k}\right]_q\). These polynomials are important combinatorial objects, often appearing in generating functions related to permutations and in representation theory. Stanley conjectured that the function \(f_{k,R}(n) = \#\left\{i : [q^{i}] \left[{n \atop k}\right]_q \equiv R \pmod{N}\right\}\) 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 \(\pi'_N(k)\) derived from the minimal period \(\pi_N(k)\) of partitions with at most \(k\) parts modulo \(N\).