AI 中文总结
研究广义停车函数多面体，证明其格切片性质并得出格点数递归，推导埃尔哈特多项式公式，在特殊情况有显式形式与生成函数，还对双参数族多面体的神奇正性分类，回答相关问题并扩展前人工作。
AI 中文摘要
对于\(\mathbf{b}=(b_1,\dots,b_n)\in\mathbb{Z}_{>0}^n\)，\(\mathbf{b}\)-停车函数是正整数序列\((\beta_1,\dots,\beta_n)\)，其非递减重排\(\beta_1'\le\beta_2'\le\cdots\le\beta_n'\)满足\(\beta_i'\le b_1+\cdots+b_i\)。\(\mathbf{b}\)-停车函数多面体\(\mathfrak{X}_n(\mathbf{b})\)是\(\mathbb{R}^n\)中长度为\(n\)的所有\(\mathbf{b}\)-停车函数的凸包。我们证明通过将一个坐标固定为整数值得到的\(\mathfrak{X}_n(\mathbf{b})\)的每个格切片本身是低一维的\(\mathbf{b}'\)-停车函数多面体，给出了\(\mathbf{b}'\)的显式参数向量，这产生了\(\mathfrak{X}_n(\mathbf{b})\)格点数的递归。我们还表明\(\mathbf{b}\)-停车函数多面体的每个伸缩是另一个此类多面体的平移，格点数是\(\mathbf{b}\)的多项式函数，并推导出\(\mathfrak{X}_n(\mathbf{b})\)任意\(\mathbf{b}\)的埃尔哈特多项式的显式公式。在\(\mathbf{b}=(a,b,\dots,b)\)的特殊情况下，得到了埃尔哈特多项式的显式封闭形式和生成函数。作为应用，我们对双参数族\(\mathfrak{X}_n(a,b)=\mathfrak{X}_n(a,b,\dots,b)\)中的神奇正性进行了分类：多面体\(\mathfrak{X}_n(a,b)\)是神奇正的当且仅当\((n,a,b)\ne(2,1,1)\)。因此，我们回答了费罗尼和东谷提出的关于\(\mathfrak{X}_n(a,b)\)的问题。我们的结果扩展了刘和张关于部分排列多面体的近期工作，并使我们推测对于\(n\ge3\)的每个\(\mathfrak{X}_n(\mathbf{b})\)神奇正性都成立。
英文摘要
For $\mathbf{b}=(b_1,\dots,b_n)\in\mathbb{Z}_{>0}^n$, a $\mathbf{b}$-parking function is a sequence $(β_1,\dots,β_n)$ of positive integers whose nondecreasing rearrangement $β_1'\leβ_2'\le\cdots\leβ_n'$ satisfies $β_i'\le b_1+\cdots+b_i$. The $\mathbf{b}$-parking-function polytope $\mathfrak{X}_n(\mathbf{b})$ is the convex hull of all $\mathbf{b}$-parking functions of length $n$ in $\mathbb{R}^n$. We prove that every lattice slice of $\mathfrak{X}_n(\mathbf{b})$, obtained by fixing one coordinate at an integer value, is itself a $\mathbf{b}'$-parking-function polytope of one dimension less, with an explicit parameter vector $\mathbf{b}'$; this yields a recursion for the number of lattice points of $\mathfrak{X}_n(\mathbf{b})$. We further show that every dilate of a $\mathbf{b}$-parking-function polytope is a translate of another such polytope, that the number of lattice points is a polynomial function of $\mathbf{b}$, and we deduce an explicit formula for the Ehrhart polynomial of $\mathfrak{X}_n(\mathbf{b})$ for arbitrary $\mathbf{b}$ as a finite sum indexed by draconian sequences, resolving a problem of Hanada, Lentfer, and Vindas-Meléndez; an equivalent formula was recently obtained, independently, by Liu and Thawinrak in a closely related setting. In the special case $\mathbf{b}=(a,b,\dots,b)$, we obtain an explicit closed form and a generating function for the Ehrhart polynomial. As an application, we classify magic positivity in the two-parameter family $\mathfrak{X}_n(a,b)=\mathfrak{X}_n(a,b,\dots,b)$: the polytope $\mathfrak{X}_n(a,b)$ is magic positive if and only if $(n,a,b)\ne(2,1,1)$. Thus, we answer a problem posed by Ferroni and Higashitani for $\mathfrak{X}_n(a,b)$. Our result extends recent work of Liu and Zhang on partial permutahedra and leads us to conjecture that magic positivity holds for every $\mathfrak{X}_n(\mathbf{b})$ with $n\ge3$.
Comments29 pages, 4 figures. We welcome comments!
