A generalization of a 1998 unimodality conjecture of Reiner and Stanton

An interesting, and still wide open, conjecture of Reiner and Stanton predicts that certain “strange” symmetric differences of $q$-binomial coefficients are always nonnegative and unimodal. We extend their conjecture to a broader, and perhaps more natural, framework, by conjecturing that, for each $k \geq 5$, the polynomials\[f(k,m,b)(q) = {{\bigl (\frac{m}{k} \bigr )}_q} - {q^{\frac{k(m-b)}{2}+b-2k+2} \cdot {{\bigl ( \frac{b}{k-2} \bigr )}_q}}\]are nonnegative and unimodal for all $m \gg {}_k 0$ and $b \leq \frac{km-4k+4}{k-2}$ such that $kb \equiv km$ (mod $2$), with the only exception of $b = \frac{km-4k+2}{k-2}$ when this is an integer.

Using the KOH theorem, we combinatorially show the case $k = 5$. In fact, we completely characterize the nonnegativity and unimodality of $f(k,m, b)$ for $k \leq 5$. (This also provides an isolated counterexample to Reiner–Stanton’s conjecture when $k = 3$.) Further, we prove that, for each $k$ and $m$, it suffices to show our conjecture for the largest $2k-6$ values of $b$.

Keywords

$q$-binomial coefficient, Gaussian polynomial, unimodality, KOH theorem, positivity

2010 Mathematics Subject Classification

Primary 05A15. Secondary 05A17, 05A19.

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The second author was partially supported by a Simons Foundation grant (#274577).

Received 27 November 2017

Published 27 September 2019