发表机构
Texas State University（德州州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出松弛共停车函数，通过纤维系统证明拟阵h-向量为纯O-序列，从而在多种拟阵类上验证Stanley猜想。
AI 中文摘要
Stanley猜想断言拟阵的$h$-向量是纯$O$-序列。Corry、Dochtermann、McClain、Perkinson和Yi引入了圈系统，通过共停车函数为允许圈系统的拟阵给出了一个双射证明，并提出了广义圈系统。每个允许圈系统的拟阵都是正则的，而Wagner图和Petersen图不允许由圈组成的广义圈系统。我们提出了一种松弛。在共停车递归失效之处，即在唯一并集为独立的层上，共停车函数被一个纤维所取代：一个纯复形，其$h$-向量为死节点（即附着于该层的拟阵的次拟阵）的$h$-向量。我们证明，对于任何具有固定基的拟阵，只要所需的纤维存在，松弛共停车函数就形成一个纯复形，其次数序列就是该拟阵的$h$-向量，因此Stanley猜想成立。证明依赖于对具有固定基的拟阵的Dhar燃烧算法的一个版本，该算法给出了纯度，以及一个删除-收缩恒等式，该恒等式将$h$-向量与共停车函数之间的差距计算为局部$h$-向量的和，每个死节点对应一个。锥化、双锥化和三锥化图及其规范生成树携带纤维系统，Wagner图和Petersen图也是如此。在图之外，秩至多为三的拟阵、余秩为二的拟阵或均匀拟阵的每个基都携带纤维系统，因此Stanley猜想对这些类成立。我们还给出了一个半径为二的十二顶点图，该图及其广度优先生成树不携带纤维系统。
英文摘要
Stanley's conjecture asserts that the $h$-vector of a matroid is a pure $O$-sequence. Corry, Dochtermann, McClain, Perkinson and Yi introduced cycle systems, which give a bijective proof for the matroids that admit one, through coparking functions, and proposed generalized cycle systems. Every matroid admitting a cycle system is regular, and the Wagner and Petersen graphs admit no generalized cycle system consisting of circuits. We propose a relaxation. Where the coparking recursion breaks down, at the strata whose unique union is independent, the coparking functions are replaced by a fibre: a pure multicomplex with the $h$-vector of the dead node, the minor of the matroid attached to the stratum. We prove, for any matroid with a fixed basis, that whenever the required fibres exist the relaxed coparking functions form a pure multicomplex whose degree sequence is the $h$-vector of the matroid, so Stanley's conjecture follows. The proof rests on a version of Dhar's burning algorithm for a matroid with a fixed basis, which gives the purity, and on a deletion-contraction identity that computes the gap between the $h$-vector and the coparking functions as a sum of local $h$-vectors, one per dead node. Coned, biconed and triconed graphs carry a fibre system with their canonical spanning trees, as do the Wagner and Petersen graphs. Beyond graphs, every basis of a matroid of corank two, of a matroid of rank at most four or of a uniform matroid carries a fibre system, so Stanley's conjecture follows for those classes. We also give a graph of radius two on twelve vertices which, with its breadth-first spanning tree, carries no fibre system.
Comments46 pages, 16 figures. v2: rank at most four added, definition of a fibre simplified
