Journal of Combinatorics

Volume 11 (2020)

Number 1

A generalization of a 1998 unimodality conjecture of Reiner and Stanton

Pages: 111 – 126



Richard P. Stanley (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Fabrizio Zanello (Department of Mathematical Sciences, Michigan Technological University, Houghton, Mich., U.S.A.)


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 $b$.


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

2010 Mathematics Subject Classification

Primary 05A15. Secondary 05A17, 05A19.

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

Received 27 November 2017

Published 27 September 2019