AI 中文总结
针对Chapoton关于树多面体对应多项式h(τ)的伽马正性猜想，本文给出章鱼形树的双射证明，并将结果扩展至偏章鱼形树。
AI 中文摘要
Chapoton引入了一类有趣的多面体，称为树多面体（arbor polytopes），其中任意树τ对应一个多面体Q_τ。Chapoton猜想，按非零元素数量计数Q_τ中格点的多项式h(τ)是回文且单峰的。Athanasiadis、Xiao和Yan最近证明，在他们称为章鱼（octopuses）的某类树中，h(τ)是伽马正性的。由于他们的证明是计算性的，他们要求给出一个双射证明。我们给出了这样的证明，还扩展了他们的结果，证明在我们称为偏章鱼（lopsided octopuses）的更大类树中，h(τ)也是伽马正性的。
英文摘要
Chapoton introduced an interesting family of polytopes called arbor polytopes, where a polytope $\mathcal{Q}_τ$ is associated to any arbor $τ$. Chapoton conjectured that the polynomial $h(τ)$ which counts lattice points in $\mathcal{Q}_τ$ by number of nonzero entries is palindromic and unimodal. Athanasiadis, Xiao, and Yan recently proved that $h(τ)$ is gamma-positive for a certain class of arbors they call octopuses. Since their proof was computational, they asked for one that is bijective. We present such a proof. We also extend their results by showing that $h(τ)$ is gamma-positive for a larger class of arbors we call lopsided octopuses.
Comments7 pages, 2 figures
