A Unified Approach to Unimodality of Gaussian Polynomials

Koutschan, Christoph; Uncu, Ali Kemal; Wong, Elaine

doi:10.1145/3597066.3597113

A Unified Approach to Unimodality of Gaussian Polynomials

Conference · 01 July 2023

DOI:https://doi.org/10.1145/3597066.3597113· OSTI ID:1994673

Koutschan, Christoph ^[1]; Uncu, Ali Kemal ^[1]; Wong, Elaine ^[2]

Johann Radon Institute for Computational and Applied Mathematics
ORNL

In 2013, Pak and Panova proved the strict unimodality property of q-binomial coefficients (as polynomials in q) based on the combinatorics of Young tableaux and the semigroup property of Kronecker coefficients. They showed it to be true for all ℓ, m ≥ 8 and a few other cases. We propose a different approach to this problem based on computer algebra, where we establish a closed form for the coefficients of these polynomials and then use cylindrical algebraic decomposition to identify exactly the range of coefficients where strict unimodality holds. This strategy allows us to tackle generalizations of the problem, e.g., to show unimodality with larger gaps or unimodality of related sequences. In particular, we present proofs of two additional cases of a conjecture by Stanley and Zanello.

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Research Organization:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Organization:: USDOE

DOE Contract Number:: AC05-00OR22725

OSTI ID:: 1994673

Country of Publication:: United States

Language:: English

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