A counterexample to a log-concavity conjecture of Brenti

Christian Gaetz

Abstract

This note records a counterexample to Brenti's 1998 conjecture that the nonzero coefficients of $\widetilde{R}$-polynomials form a log-concave sequence.

Disclosure

“e (shortening u and v). Both counterexamples were verified by the author using the coxeter3 package in SageMath [Sage24]. The paper was written by the author, with minor editing carried out by AI tools. 1While writing this article, the author learned of another recently discovered counterexample in type H 4 [Pau26].”
PDF page 2

Classification: Substantial mathematical content or result generation
Multiplier: 10
Verified: 2026-08-23T05:14:35.269236Z

Structural counts

Pages 3 pdf

Theorems 1 source

Lemmas 0 source

Propositions 0 source

Corollaries 0 source

Definitions 0 source

Displayed equations 6 source

Bibliography entries 22 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file main.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.