A combinatorial proof of strict unimodality for q-binomial coefficients

I. Pak and G. Panova recently proved that the g-binomial coefficient ((m+n)(m))(q) 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. (C) 2014 Elsevier B.V. All rights reserved.