S-meandric 排列与切触多项式
S-meandric Permutations and Tangency Polynomials
浏览论文内容
中文总结 AI 辅助
本文研究允许切触时 meander 配置与排列的非唯一对应，给出可实现性判据，证明实现构成仿射空间，引入切触多项式并证明其分解性质，推导中心极限定理与渐近公式。
中文摘要 AI 辅助
一个 meander 是由两条简单平面曲线横截相交构成的配置。其交点的顺序定义了一个排列，该排列决定了配置。然而，当允许切触时，不同的配置可能共享同一个排列。我们研究了由这种非唯一性产生的组合与代数结构。我们给出了一个可实现性判据，并证明了每个可实现排列的实现构成二元域上的一个仿射空间。我们使用一个称为分量脊柱的关联图来描述该空间。我们证明了每个排列的分量脊柱都是仙人掌图。我们还引入了切触多项式，它按切触次数计数实现，研究了其性质，并证明了它在分量脊柱的环和桥上分解。我们推导了切触次数的中心极限定理，并获得了不同切触多项式数量的渐近公式。
英文摘要
A meander is a configuration of two simple plane curves intersecting transversely. The orders of their intersection points define a permutation that determines the configuration. When tangencies are allowed, however, different configurations can share the same permutation. We study the combinatorial and algebraic structures arising from this non-uniqueness. We give a realization criterion and show that the realizations of each realizable permutation form an affine space over the two-element field. We describe this space using an associated graph, called the component spine. We prove that the component spine of every permutation is a cactus. We also introduce the tangency polynomial, which counts realizations by their number of tangencies, investigate its properties, and prove that it factors over the cycles and bridges of the component spine. We derive a central limit theorem for tangency counts and obtain asymptotic formulas for the number of distinct tangency polynomials.
发表机构
- Leonhard Euler International Mathematical Institute in Saint Petersburg（圣彼得堡列昂哈德·欧拉国际数学研究所）
- Saint Petersburg State University（圣彼得堡国立大学）
机构由 AI 辅助整理，请以论文原文为准。
