The ultra log-concavity of Z-polynomials and γ-polynomials of uniform matroids

Abstract

Proudfoot, Xu, and Young introduced the Z-polynomial for any matroid and conjectured the polynomial has only real roots. Recently, Ferroni, Nasr, and Vecchi introduced the γ-polynomial of a matroid. In this paper, we prove that both the Z-polynomials and γ-polynomials of uniform matroids are ultra log-concave, which due to Newton's inequality partially supports the real-rootedness conjecture. We also give an alternative formula for the γ-polynomials of uniform matroids. As an application, we use this formula to provide a new proof of the γ-positivity of sparse paving matroids.

Keywords:

• Log-concavity
• Z-polynomial
• γ-polynomial
• uniform matroid
• sparse paving matroid
• recurrence relation

2020 AMS SUBJECT CLASSIFICATIONS:

• 05A15
• 05B35
• 33F10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The HolonomicFunctions package can be downloaded at https://www3.risc.jku.at/research/combinat/software/ergosum/RISC/HolonomicFunctions.html.

Additional information

Funding

Xie is supported by the National Natural Science Foundation of China under Grant No. 12271403.   Zhang is supported by the National Natural Science Foundation of China under Grant No. 12171362.