Research by Charles Brittenham, Andrew Carroll (faculty), T. Kyle Petersen (faculty) , and Connor Thomas
We attempted to find combinatorial proofs of unimodality for various number sets, namely the q-analogue of n!, the q-binomial coefficients, and integer partitions with distinct parts of size at most n. We proceeded by attempting to find a sign-reversing involution on the gamma-vector expansions for each of these polynomials to show that the entries of these vectors were nonnegative, and hence that the polynomials modeled by those gamma-vectors are unimodal. While we were able to show this for the q-analogue of n!, further refining of the involutions for the remaining two number sets is needed to give a complete proof of unimodality.
Published as: Unimodality via alternating gamma vectors, Electron. J. Combin. 23 (2016), no. 2, Paper 2.40, 22 pp.