A combinatorial proof of strict unimodality for [formula omitted]-binomial coefficients

Abstract

I. Pak and G. Panova recently proved that the q-binomial coefficient m+nmq is a strictly unimodal polynomial in q for m,n≥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≥8d, and 2d≤r≤mn/2, then the rth rank of L(m,n) has at least d more elements than the next lower rank.