A combinatorial proof of strict unimodality for -binomial coefficients

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. 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 ≥ 8 d , and 2 d ≤ r ≤ m n / 2 , then the r th rank of L ( m , n ) has at least d more elements than the next lower rank.