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

Abstract

Pak and Panova recently proved that the $q$-binomial coefficient ${\left(\genfrac{}{}{0ex}{}{m+n}{m}\right)}_q$ is a strictly unimodal polynomial in $q$ for $m,n\ge 8$, 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 $m,n\ge 8d$, and $2d\le r\le mn/2$, then the $r$-th rank of $L(m,n)$ has at least $d$ more elements that the next lower rank.

7 pages, expanded results

Cited by in corpus (1)

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