An n-dimensional lattice polytope <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathcal {Q}}_\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">Q</mi> <mi>σ</mi> </msub> </math></EquationSource> </InlineEquation> can be associated to any composition <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> of a positive integer n,  as a special case of constructions due to Pitman–Stanley and Chapoton. The entries of the h-vector of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\sigma ,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> introduced by Chapoton, enumerate the lattice points in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\mathcal {Q}}_\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">Q</mi> <mi>σ</mi> </msub> </math></EquationSource> </InlineEquation> by the number of their nonzero coordinates. Chapoton conjectured that this vector is equal to the h-vector of a flag simplicial polytope. This paper proves this conjecture. Moreover, it shows that the gamma-vector associated to the h-vector of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> is nonnegative by means of an explicit combinatorial interpretation and confirms certain other conjectures of Chapoton on the lattice point enumeration of composition polytopes. A combinatorial interpretation of their <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$h^*$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>h</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-polynomials is deduced.

Lattice Point Enumeration of Polytopes Associated to Integer Compositions

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2026年3月6日

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Christos A. Athanasiadis

摘要

An n-dimensional lattice polytope ${\mathcal {Q}}_\sigma $ $Q_{σ}$ can be associated to any composition $\sigma $ $σ$ of a positive integer n, as a special case of constructions due to Pitman–Stanley and Chapoton. The entries of the h-vector of $\sigma ,$ $σ,$ introduced by Chapoton, enumerate the lattice points in ${\mathcal {Q}}_\sigma $ $Q_{σ}$ 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 $ $σ$ 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 $h^*$ $h^{*}$ -polynomials is deduced.