Out of the parking lot and into the forest: Parking functions, bond lattices, and unimodal forests

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2025-06-20 DOI:10.1016/j.disc.2025.114646

Josephine Brooks , Susanna Fishel , Max Hlavacek , Sophie Rubenfeld , Bianca Carmelita Teves

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引用次数: 0

Abstract

Rota introduced the bond lattice of a graph in [11]. It's a sublattice of the set partition lattice. For certain graphs, such as triangulation graphs, it's a sublattice of the important and oft studied noncrossing partition lattice. Parking functions are another central object in algebraic combinatorics. Stanley made the connection between them by defining a bijection from maximal chains of the noncrossing partition lattice to parking functions [14]. Motivated by Stanley's bijection, we study the maximal chains in the bond lattices of triangulation graphs.

The number of maximal chains in the bond lattice of a triangulation graph is the number of ordered cycle decompositions [1], as well as being the number of rooted unimodal forests [2]. In this paper, we find a recursive bijection between these maximal chains and rooted unimodal forests, based on a simpler recursion than that given in [1].

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走出停车场，进入森林：停车功能、键格和单峰森林

Rota在[11]中介绍了图的键格。它是集合划分格的一个子格。对于某些图，如三角剖分图，它是重要且经常被研究的非交叉分割格的子格。停放函数是代数组合学中的另一个中心对象。Stanley通过定义一个从非交叉分割格的极大链到停车函数[14]的双射，建立了它们之间的联系。在Stanley双射的激励下，我们研究了三角图键格中的极大链。三角图键格中最大链的个数为有序循环分解的个数[1]，同时也为有根单峰森林的个数[2]。在本文中，我们基于一个比[1]中给出的更简单的递归，找到了这些极大链和有根单峰森林之间的递归双射。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.