Enumerating neighborly polytopes and oriented matroids
Enumerating neighborly polytopes and oriented matroids
复制标题
枚举邻近多面体和定向拟阵
DOI：
10.1080/10586458.2015.1015084
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发表时间：
2015
影响因子：
0.5
通讯作者：
Hiroyuki Miyata and Arnau Padrol
中科院分区：
文献类型：
--
作者：
横山信彦;松本嘉寛;飯田圭一郎;薛宇孝;遠藤誠;中島康晴;大庭伸介;Hiroyuki Miyata and Arnau Padrol
邻接多面体是指在具有相同顶点数的所有多面体中最大化每个维度中的面数的多面体。尽管它们具有极端的性质，但它们形成了一类令人惊讶的丰富的多面体，这一类已被广泛研究，并成为许多悬而未决的问题和猜想的主题。在这篇文章中，我们研究了邻域多面体的计数，超越了到目前为止已经计算过的情况。为此，我们列举了邻域定向拟阵--邻域多面体的组合抽象--小秩次和小序数。特别地，如果我们用OM(r，n)表示秩元的所有定向拟阵的集合，我们就确定了OM(5，≤12)、OM(6，≤9)、OM(7，≤11)和OM(9，≤12)中的所有一致邻域定向拟阵以及OM(6，10)和OM(8，11)中所有可能的邻域定向拟阵的面格。此外，我们还对OM(7，10)和OM(8，11)中一致2邻域定向拟阵的所有可能面格进行了分类。在此基础上，我们构造了许多有趣的例子，并对开放猜想进行了检验。
Neighborly polytopes are those that maximize the number of faces in each dimension among all polytopes with the same number of vertices. Despite their extremal properties, they form a surprisingly rich class of polytopes, which has been widely studied and is the subject of many open problems and conjectures. In this article, we study the enumeration of neighborly polytopes beyond the cases that have been computed so far. To this end, we enumerate neighborly oriented matroids—a combinatorial abstraction of neighborly polytopes—of small rank and corank. In particular, if we denote by OM(r,n) the set of all oriented matroids of rankrandnelements, we determine all uniform neighborly oriented matroids in OM(5, ≤12), OM(6, ≤9), OM(7, ≤11), and OM(9, ≤12) and all possible face lattices of neighborly oriented matroids in OM(6, 10) and OM(8, 11). Moreover, we classify all possible face lattices of uniform 2-neighborly oriented matroids in OM(7, 10) and OM(8, 11). Based on the enumeration, we construct many interesting examples and test open conjectures.