(132,213)-回避置换多面体的 Ehrhart $h^*$-多项式：一个修复锥与最终实根性
Ehrhart $h^*$-polynomials of $(132,213)$-avoiding permutation polytopes: A repair cone and eventual real-rootedness
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中文总结 AI 辅助
本研究通过构造修复锥和几何尾部不等式，证明了(132,213)-回避置换多面体的Ehrhart h*-多项式在特定范围内具有实根性，并验证了体积猜想。
中文摘要 AI 辅助
设 $P_d(132,213)$ 为 $S_d$ 中回避 $132$ 和 $213$ 的置换的凸包，并设 $H_d(t)$ 为其 Ehrhart $h^*$-多项式，由 $\sum_{m\ge0}|mP_d(132,213)\cap\mathbb{Z}^d|t^m=\frac{H_d(t)}{(1-t)^d}$ 定义。我们给出了该多面体、一个路径-拉普拉斯亏损多面体以及两倍平移链-偏序集置换多胞体之间的显式格等价。由此得到的面描述和三角剖分给出了 Davis 和 Sagan 的立方性与体积猜想的一个自包含证明，其归一化体积为 $2^{d-1}d^{d-3}$（$d\ge2$）。对于 $d\ge3$，$H_d(t)$ 中 $t^{d-1}$ 的系数计数强锦标赛得分序列。$H_d(t)$ 的精细欧拉展开在 $d=7$ 时已出现负坐标。我们构造了一个相容的相邻符号差分锥以容纳此障碍。其最大均匀修复参数为 $2-\sqrt{3}$，且成员资格归结为显式的几何尾部不等式。一个精确数据集和重构算法验证了这些不等式直到 $d=1000$。一个标记分量恒等式、离散平滑和 Darroch 众数定理（Ann. Math. Statist. 35 (1964), 1317--1321, 定理 4）对所有 $d\ge2^{72}$ 证明了它们。因此，$H_d(t)$ 在 $3\le d\le1000$ 和 $d\ge2^{72}$ 时仅有负实零点，而 $H_1(t)=H_2(t)=1$。在中间范围内的均匀实根性仍然开放。
英文摘要
Let $P_d(132,213)$ be the convex hull of the permutations in $S_d$ that avoid $132$ and $213$, and let $H_d(t)$ be its Ehrhart $h^*$-polynomial, defined by $\sum_{m\ge0}|mP_d(132,213)\cap\mathbb{Z}^d|t^m=\frac{H_d(t)}{(1-t)^d}$. We give an explicit lattice equivalence between this polytope, a path-Laplacian deficit polytope, and twice a translated chain-poset permutahedron. The resulting face description and triangulation give a self-contained proof of Davis and Sagan's cubicality and volume conjecture, with normalized volume $2^{d-1}d^{d-3}$ for $d\ge2$. For $d\ge3$, the coefficient of $t^{d-1}$ in $H_d(t)$ counts strong tournament score sequences. The refined Eulerian expansion of $H_d(t)$ has a negative coordinate already at $d=7$. We construct a compatible cone of adjacent signed differences that accommodates this obstruction. Its largest uniform repair parameter is $2-\sqrt{3}$, and membership reduces to explicit geometric-tail inequalities. An exact dataset and reconstruction algorithm verify these inequalities through $d=1000$. A marked-component identity, discrete smoothing, and Darroch's mode theorem (Ann. Math. Statist. 35 (1964), 1317--1321, Theorem 4) prove them for every $d\ge2^{72}$. Thus $H_d(t)$ has only negative real zeros for $3\le d\le1000$ and for $d\ge2^{72}$, while $H_1(t)=H_2(t)=1$. Uniform real-rootedness in the intervening range remains open.
发表机构
- Centro de Informática, Universidade Federal de Pernambuco（伯南布哥联邦大学信息学中心）
机构由 AI 辅助整理，请以论文原文为准。
