Coefficients of Gaussian Polynomials Modulo $N$

TL;DR

This paper proves that the distribution of coefficients of Gaussian polynomials modulo any integer N is quasipolynomial, extending Stanley's conjecture from N=2 to all positive integers, and identifies their quasi-periods.

Contribution

It generalizes Stanley's conjecture, showing the coefficients' distribution modulo N is quasipolynomial for all N, and determines the associated quasi-periods.

Findings

Distribution of Gaussian polynomial coefficients modulo N is quasipolynomial.

The quasi-period is derived from the minimal period of partitions modulo N.

The result extends Stanley's conjecture from N=2 to all positive integers.

Abstract

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 $N$.