Inequalities between gamma-polynomials of graph-associahedra

TL;DR

This paper proves a conjecture relating to inequalities between gamma-polynomials of graph-associahedra by introducing a partial order on tree graphs and analyzing the effects of specific graph operations.

Contribution

It defines a new partial order on trees that induces inequalities between gamma-polynomials and extends the analysis to non-tree graphs using known moves.

Findings

Partial order on trees induces gamma-polynomial inequalities

Tree shifts lower gamma-polynomials of trees

Flossing moves lower gamma-polynomials of non-tree graphs

Abstract

We prove a conjecture of Postnikov, Reiner and Williams by defining a partial order on the set of tree graphs with $n$ vertices that induces inequalities between the $\gamma$-polynomials of their associated graph-associahedra. The partial order is given by relating trees that can be obtained from one another by operations called tree shifts. We also show that tree shifts lower the $\gamma$-polynomials of graphs that are not trees, as do the flossing moves of Babson and Reiner.