Frankl-F\"{u}redi-Kalai Inequalities on the $\gamma$-vectors of flag nestohedra

TL;DR

This paper introduces a new simplicial complex associated with flag nestohedra, proving that their gamma-vectors satisfy Frankl-F"uredi-Kalai inequalities, advancing understanding of their combinatorial properties.

Contribution

It constructs a flag simplicial complex for flag nestohedra and proves the gamma-vector satisfies Frankl-F"uredi-Kalai inequalities, partially resolving a conjecture.

Findings

Gamma-vectors of flag nestohedra satisfy Frankl-F"uredi-Kalai inequalities

Construction of a new flag simplicial complex related to nestohedra

Comparison with existing complexes for specific flag nestohedra

Abstract

For any flag nestohedron, we define a flag simplicial complex whose $f$-vector is the $\gamma$-vector of the nestohedron. This proves that the $\gamma$-vector of any flag nestohedron satisfies the Frankl-F\"{u}redi-Kalai inequalities, partially solving a conjecture by Nevo and Petersen \cite{np}. We also compare these complexes to those defined by Nevo and Petersen in \cite{np} for particular flag nestohedra.