论文标题
某些图，标准单体和球形停车功能的骨架理想
Skeleton Ideals of Certain Graphs, Standard Monomials and Spherical Parking Functions
论文作者
论文摘要
让$ g $为顶点集合的（定向）图$ v = \ {0，1，\ ldots，n \} $，带有root $ 0 $。 Postnikov和Shapiro关联了单个理想$ \ MATHCAL {M} _g $在多项式环$ r = {\ Mathbb {k}} [x_1，\ ldots，x_n] $上的字段$ \ Mathbb {k {k} $。 $ \ Mathcal {（k）} $ $ \ Mathcal {M} $ subideal $ \ Mathcal {M} _g^{（k）} $由$ \ widetilde {v} = v \ setminus \ \ setminus \ {0 \} $ a $ k+1 $的$ k+k $ k $ skelet gextion a $ k $ $ skelet n offection of $ \ setMinus \ {0 \} $的子集生成。 Dochtermann为某些类似的简单图形$ g $获得了许多有趣的同源和组合属性$ 1 $ -Skeleton的理想$ \ Mathcal {M} _G^{（1）} $。有限序列$ \ MATHCAL {p} =（p_1，\ ldots，p_n）\ in \ mathbb {n}^n $称为球形$ g $ -parking函数，如果单元$ \ mathbf {x}} x_i^{p_i} \ in \ mathcal {m} _g \ setMinus \ mathcal {m} _g^{（n-2）} $。令$ {\ rm spf}（g）$为所有球形$ g $ parking函数的集合。在本文中，给出了$ k $ - 骨骼的所有多层betti数字的组合描述。另外，使用珀金森 - 杨（用于简单图）和Gaydarov-hopkins（用于多机）的DFS燃烧算法，我们对图形的球形$ g $ parking函数进行组合解释，用于图$ g = k = k = k_ {n+1} $ n exder n+n exder n+n+n+n+n. $ e $。特别是，我们证明了$ | {\ rm spf}（k_ {n+1} - \ {e_0 \}）| =（n-1）^{n-1} $ for Edge $ e_0 $通过root $ 0 $ 0 $，但是=（n-1）^{n-3}（n-2）^2 $对于边缘$ e_1 $而不是通过根。
Let $G$ be an (oriented) graph on the vertex set $V = \{ 0, 1,\ldots,n\}$ with root $0$. Postnikov and Shapiro associated a monomial ideal $\mathcal{M}_G$ in the polynomial ring $ R = {\mathbb{K}}[x_1,\ldots,x_n]$ over a field $\mathbb{K}$. A subideal $\mathcal{M}_G^{(k)}$ of $\mathcal{M}_G$ generated by subsets of $\widetilde{V}=V\setminus \{0\}$ of size at most $k+1$ is called a $k$-skeleton ideal of the graph $G$. Many interesting homological and combinatorial properties of $1$-skeleton ideal $\mathcal{M}_G^{(1)}$ are obtained by Dochtermann for certain classes of simple graph $G$. A finite sequence $\mathcal{P}=(p_1,\ldots,p_n) \in \mathbb{N}^n$ is called a spherical $G$-parking function if the monomial $\mathbf{x}^{\mathcal{P}} = \prod_{i=1}^{n} x_i^{p_i} \in \mathcal{M}_G \setminus \mathcal{M}_G^{(n-2)}$. Let ${\rm sPF}(G)$ be the set of all spherical $G$-parking functions. In this paper, a combinatorial description for all multigraded Betti numbers of the $k$-skeleton ideal $\mathcal{M}_{K_{n+1}}^{(k)}$ of the complete graph $K_{n+1}$ on $V$ are given. Also, using DFS burning algorithms of Perkinson-Yang-Yu (for simple graph) and Gaydarov-Hopkins (for multigraph), we give a combinatorial interpretation of spherical $G$-parking functions for the graph $G = K_{n+1}- \{e\}$ obtained from the complete graph $K_{n+1}$ on deleting an edge $e$. In particular, we showed that $|{\rm sPF}(K_{n+1}- \{e_0\} )|= (n-1)^{n-1}$ for an edge $e_0$ through the root $0$, but $|{\rm sPF}(K_{n+1} - \{e_1\})| = (n-1)^{n-3}(n-2)^2$ for an edge $e_1$ not through the root.
