Unimodality via alternating gamma vectors

Abstract

For a polynomial with palindromic coefficients, unimodality is equivalent to having a nonnegative g-vector. A sufficient condition for unimodality is having a non-negative γ-vector, though one can have negative entries in the γ-vector and still have a nonnegative g-vector. In this paper we provide combinatorial models for three families of γ-vectors that alternate in sign. In each case, the γ-vectors come from unimodal polynomials with straightforward combinatorial descriptions, but for which there is no straightforward combinatorial proof of unimodality. By using the transformation from γ-vector to g-vector, we express the entries of the g-vector combinatorially, but as an alternating sum. In the case of the q-analogue of n!, we use a sign-reversing involution to interpret the alternating sum, resulting in a manifestly positive formula for the g-vector. In other words, we give a combinatorial proof of unimodality. We consider this a “proof of concept” result that we hope can inspire a similar result for the other two cases, ∏nj=1(1 + qj) and the q-binomial coefficient [nk].