单位区间图上的峰与峰嵌套
Peaks and peak-nestings on unit interval graphs
- Stockholm University（斯德哥尔摩大学）
- KTH Royal Institute of Technology（皇家理工学院）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文在单位区间图上引入峰嵌套统计量，通过Dyck路径与双色Motzkin路径的双射细化Touchard恒等式，研究各类子图的多项式性质，并联系格路径拟阵。
中文摘要 AI 辅助
单位区间图通过面积序列，或等价地通过Dyck路径，承认经典的Catalan编码。我们在这些图上引入一个新的统计量，称为峰嵌套，定义为包含公共顶点的峰团的最大数量。通过与面积序列的对应关系，峰嵌套也定义了Dyck路径上的一个新统计量。我们在Dyck路径和双色Motzkin路径之间构造了一个双射，该双射同时记录峰嵌套和峰的数量。这产生了Touchard恒等式的一个细化，涉及给定高度的Dyck路径的系数公式，以及OEIS中记录的若干序列的新组合解释。我们研究了所有单位区间图以及连通、阿贝尔、$2$-嵌套和约化单位区间图子类上的峰和峰嵌套多项式。对于这些族，我们获得了闭式公式、递推关系，并且对于对称情形，在伽马基中获得了具有显式组合解释的非负展开。我们还研究了关于相应系数序列的零点位置和对数凹性的问题：一些多项式族形成广义Sturm序列，而其他多项式族则不是实根的，但似乎具有对数凹系数。最后，我们将单位区间图的峰团表示与格路径拟阵联系起来。
英文摘要
Unit interval graphs admit a classical Catalan encoding by area sequences, or equivalently by Dyck paths. We introduce a new statistic on these graphs, called peak-nesting, defined as the largest number of peak-cliques containing a common vertex. Through the correspondence with area sequences, peak-nesting also defines a new statistic on Dyck paths. We construct a bijection between Dyck paths and bicolored Motzkin paths which simultaneously records peak-nesting and the number of peaks. This yields a refinement of Touchard's identity, coefficient formulas involving Dyck paths of given height, and new combinatorial interpretations for several sequences recorded in the OEIS. We study the peak and peak-nesting polynomials over all unit interval graphs and over the subclasses of connected, Abelian, $2$-nested, and reduced unit interval graphs. For these families we obtain closed formulas, recurrences, and, for the symmetric cases, nonnegative expansions in the gamma-basis with explicit combinatorial interpretations. We also investigate questions regarding the location of zeros and the log-concavity of the corresponding coefficient sequences: some of the polynomial families form generalized Sturm sequences, whereas others fail to be real-rooted but appear nevertheless to have log-concave coefficients. Finally, we relate the peak-clique presentation of a unit interval graph to lattice path matroids.
