A combinatorial proof of strict unimodality for q-binomial coefficients

摘要

I. Pak and G. Panova recently proved that the q-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.Wegive 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.