The $\gamma$-vector of a barycentric subdivision

TL;DR

This paper proves that the gamma-vector of a barycentric subdivision of a simplicial sphere corresponds to the f-vector of a balanced simplicial complex, linking combinatorial invariants through refined Eulerian numbers.

Contribution

It establishes a new combinatorial interpretation of the gamma-vector for barycentric subdivisions, connecting it to balanced simplicial complexes.

Findings

Gamma-vector equals the f-vector of a balanced simplicial complex.

Refined Eulerian numbers are used to analyze the h-vector.

Provides a combinatorial basis for gamma-vector properties.

Abstract

We prove that the $\gamma$-vector of the barycentric subdivision of a simplicial sphere is the $f$-vector of a balanced simplicial complex. The combinatorial basis for this work is the study of certain refinements of Eulerian numbers used by Brenti and Welker to describe the $h$-vector of the barycentric subdivision of a boolean complex.