A combinatorial proof of strict unimodality for qq-binomial coefficients

I. Pak and G. Panova recently proved that the qq-binomial coefficient m+nmq is a strictly unimodal polynomial in qq for m,n≥8m,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)L(m,n). In fact, we prove a stronger result: if m,n≥8dm,n≥8d, and 2d≤r≤mn/22d≤r≤mn/2, then the rrth rank of L(m,n)L(m,n) has at least dd more elements than the next lower rank.