A positivity conjecture for a quotient of $q$-binomial coefficients

TL;DR

This paper proposes a conjecture that certain quotients of $q$-binomial coefficients have non-negative coefficients, proves it in specific cases, and extends it to fake Gaussian sequences, with implications for cyclic sieving polynomials.

Contribution

It introduces a new positivity conjecture for quotients of $q$-binomial coefficients, proves it in two cases, and extends the conjecture to fake Gaussian sequences.

Findings

Proved the conjecture in two non-trivial cases.

Extended the conjecture to D. Stanton's fake Gaussian sequences.

Showed that certain cyclic sieving polynomials have non-negative coefficients.

Abstract

We conjecture that, if the quotient of two $q$-binomial coefficients with the same top argument is a polynomial, then it has non-negative coefficients. We summarise what is known about the conjecture and prove it in two non-trivial cases. Moreover, we move ahead to extend our conjecture to D. Stanton's fake Gaussian sequences. As a corollary we obtain that a polynomial that is conjectured to be a cyclic sieving polynomial for Kreweras words [S. Hopkins and M. Rubey, Selecta Math. (N.S.) 28 (2022), Paper No. 10] is indeed a polynomial with non-negative integer coefficients.