Lattice point enumeration of polytopes associated to integer compositions
Christos A. Athanasiadis
TL;DR
This work studies lattice-point enumeration for composition polytopes ${\mathcal Q}_{\sigma}$ associated to a composition $\sigma$ of $n$. It proves that the $h$-vector $h(\sigma)=(h_0(\sigma),\dots,h_n(\sigma))$ is $\gamma$-positive, hence palindromic and unimodal, by giving a concrete combinatorial interpretation of the $\gamma_i(\sigma)$ coefficients; it also establishes a tight link between Ehrhart theory and zeta polynomials via ${\rm Ehr}({\mathcal Q}_{\sigma},m)={\mathcal Z}(P_{\mathrm{rev}(\sigma)},m+1)$. The paper provides two proofs of this Ehrhart–zeta relation and furnishes a combinatorial interpretation of the $h^*$-polynomial through EL-shellability of a suitable poset, showing $h^*({\mathcal Q}_{\sigma},t)=h(\Delta(P_{\mathrm{rev}(\sigma)}),t)$. Together, these results confirm Chapoton’s conjectures for composition polytopes, connect lattice-point enumeration with poset topology, and offer explicit gamma- and h*-polynomial interpretations with potential extensions to broader arbor/pitman–Stanley polytopes.
Abstract
An $n$-dimensional lattice polytope ${\mathcal Q}_σ$ can be associated to any composition $σ$ of a positive integer $n$, as a special case of constructions due to Pitman--Stanley and Chapoton. The entries of the $h$-vector of $σ$, introduced by Chapoton, enumerate the lattice points in ${\mathcal Q}_σ$ by the number of their zero coordinates. Chapoton conjectured that this vector is equal to the $h$-vector of a simplicial polytope. This paper proves that the $h$-vector of $σ$ is gamma-positive, hence palindromic and unimodal, 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.