A combinatorial proof of strict unimodality for $q$-binomial coefficients

Abstract

Pak and Panova recently proved that the $q$-binomial coefficient ${m+n \choose m}_q$ is a strictly unimodal polynomial in $q$ for $m,n \geq 8$, via the representation theory of the symmetric group. We give a direct combinatorial proof of their result by characterizing when a product of chains is strictly unimodal and then applying O'Hara's structure theorem for the partition lattice $L(m,n)$. In fact, we prove a stronger result: if $m, n \geq 8d$, and $2d \leq r \leq mn/2$, then the $r$-th rank of $L(m,n)$ has at least $d$ more elements that the next lower rank.