Proof of Almkvist's conjecture on the unimodality of partition polynomials

Abstract

For integers $r\ge 2$ and $n\ge 1$, let $$F_{r,n}(q)=\prod _{k=1}^n\frac{1-q^{rk}}{1-q^k}.$$ The coefficient of $q^j$ in \(F_{r,n}(q)\) counts partitions of $j$ into parts at most $n$, each occurring at most $r-1$ times. Hughes proved that $F_{2,n}(q)={\prod }_{k=1}^n(1+q^k)$ is unimodal for every $n\ge 1$. This result was reproved by Stanley using an algebraic approach and Odlyzko and Richmond using an analytic approach. Almkvist conjectured that $F_{r,n}(q)$ is unimodal in the following two cases: every even $r$ and every $n\ge 1$; every odd $r$ and every $n\ge 11$. He proved that this conjecture is true for $3\le r\le 20$ and $r=100,101$. In this paper, we completely settle the conjecture.

44 pages