Lattice point enumeration of polytopes associated to integer compositions

TL;DR

This paper proves Chapoton's conjecture that the $h$-vector of lattice polytopes associated with compositions matches that of a flag simplicial polytope, and confirms related conjectures with combinatorial interpretations.

Contribution

It establishes the equality of the $h$-vector with that of a flag simplicial polytope and provides combinatorial insights into the gamma-vector and $h^ ext{*}$-polynomials.

Findings

Proves Chapoton's conjecture on the $h$-vector of composition polytopes.

Shows the gamma-vector is nonnegative with an explicit combinatorial interpretation.

Provides a combinatorial interpretation of the $h^ ext{*}$-polynomials.

Abstract

An $n$-dimensional lattice polytope ${\mathcal Q}_\sigma$ can be associated to any composition $\sigma$ of a positive integer $n$, as a special case of constructions due to Pitman--Stanley and Chapoton. The entries of the $h$-vector of $\sigma$, introduced by Chapoton, enumerate the lattice points in ${\mathcal Q}_\sigma$ by the number of their nonzero coordinates. Chapoton conjectured that this vector is equal to the $h$-vector of a flag simplicial polytope. This paper proves this conjecture. Moreover, it shows that the gamma-vector associated to the $h$-vector of $\sigma$ is nonnegative by means of an explicit combinatorial interpretation and confirms certain other conjectures of Chapoton on the lattice point enumeration of composition polytopes. A combinatorial interpretation of their $h^\ast$-polynomials is deduced.