Lattice point enumeration of polytopes associated to integer compositions

Abstract

An $n$-dimensional lattice polytope ${\mathcal{Q}}_{\ddot{I}}ƒ$ can be associated to any composition $\ddot{I}ƒ$ of a positive integer $n$, as a special case of constructions due to Pitman--Stanley and Chapoton. The entries of the $h$-vector of $\ddot{I}ƒ$, introduced by Chapoton, enumerate the lattice points in ${\mathcal{Q}}_{\ddot{I}}ƒ$ 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 $\ddot{I}ƒ$ 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.

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