# Local $h$-vectors of Quasi-Geometric and Barycentric Subdivisions

## TL;DR

This paper characterizes all local h-vectors of quasi-geometric subdivisions of a simplex and proves nonnegativity of the local gamma-vector for barycentric subdivisions, advancing understanding of subdivision invariants.

## Contribution

It provides a complete characterization of local h-vectors for quasi-geometric subdivisions and establishes nonnegativity of local gamma-vectors for barycentric subdivisions.

## Key findings

- Characterization of all possible local h-vectors for quasi-geometric subdivisions
- Proof of nonnegativity of local gamma-vectors for barycentric subdivisions
- Derivation of a new recurrence formula for derangement polynomials

## Abstract

In this paper, we answer two questions on local $h$-vectors, which were asked by Athanasiadis. First, we characterize all possible local $h$-vectors of quasi-geometric subdivisions of a simplex. Second, we prove that the local $\gamma$-vector of the barycentric subdivision of any CW-regular subdivision of a simplex is nonnegative. Along the way, we derive a new recurrence formula for the derangement polynomials.