Refined Enumeration of ${{\varvec{k}}}$-plane Trees and ${\varvec{k}}$-noncrossing Trees

IF 0.6 4区数学 Q4 MATHEMATICS, APPLIED

Annals of Combinatorics Pub Date : 2023-05-10 DOI:10.1007/s00026-023-00642-6

Isaac Owino Okoth, Stephan Wagner

引用次数: 0

Abstract

A k-plane tree is a plane tree whose vertices are assigned labels between 1 and k in such a way that the sum of the labels along any edge is no greater than $k+1$. These trees are known to be related to $(k+1)$-ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting formula, where we count trees with a prescribed number of labels of each kind. Several corollaries are derived from this formula, and an analogous theorem is proven for k-noncrossing trees, a similarly defined family of labelled noncrossing trees that are related to $(2k+1)$-ary trees.

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改进了$${{\varvec{k}}}$$ k -平面树的枚举方法 $${\varvec{k}}$$

k 平面树是一种平面树，它的顶点被分配的标签介于 1 和 k 之间，使得任何一条边上的标签之和都不大于 $k+1$。众所周知，这些树与（(k+1)\）ary 树有关，而且它们是用广义版的加泰罗尼亚数来计数的。我们证明了一个简单得令人吃惊的精炼计数公式，在这个公式中，我们对每一种树都有规定数量的标签进行计数。我们从这个公式中推导出了几个推论，并证明了 k-noncrossing 树的类似定理，这是一个与 \((2k+1)\ary 树相关的有标签的非交叉树的类似定义族。

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

Annals of Combinatorics 数学-应用数学

CiteScore

1.00

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board. The scope of Annals of Combinatorics is covered by the following three tracks: Algebraic Combinatorics: Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices Analytic and Algorithmic Combinatorics: Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms Graphs and Matroids: Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches

Refined Enumeration of \({{\varvec{k}}}\)-plane Trees and \({\varvec{k}}\)-noncrossing Trees

Abstract