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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2510.23903 |
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| _version_ | 1866912844957614080 |
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| author | Athanasiadis, Christos A. |
| author_facet | Athanasiadis, Christos A. |
| contents | 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 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 $σ$ 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 $h^\ast$-polynomials is deduced. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23903 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lattice point enumeration of polytopes associated to integer compositions Athanasiadis, Christos A. Combinatorics 52B20 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 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 $σ$ 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 $h^\ast$-polynomials is deduced. |
| title | Lattice point enumeration of polytopes associated to integer compositions |
| topic | Combinatorics 52B20 |
| url | https://arxiv.org/abs/2510.23903 |