A combinatorial proof of strict unimodality for q-binomial coefficients

2014/02/05 by Vivek Dhand, Dhand, Vivek · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.1402.1199

7 pages, expanded results

arxiv created 2014/03/09 · arxiv updated 2014/03/11

Abstract

Pak and Panova recently proved that the q-binomial coefficient m+n \choose 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 r-th rank of L(m,n) has at least d more elements that the next lower rank.

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