Tree statistics from Matula numbers
Emeric Deutsch
在自然数和有根树之间存在一一对应的关系;这个数被称为有根树的Matula数。我们展示了如何大量的属性的树木可以直接从相应的Matula数。
There is a one-to-one correspondence between natural numbers and rooted trees; the number is called the Matula number of the rooted tree. We show how a large number of properties of trees can be obtained directly from the corresponding Matula number.
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Three remarks on Matula numbers
在SIAM Review 10，第273页，D. W. Matula描述了N和拓扑有根树的集合之间的一个双射，这个数称为有根树的Matula数。Gutman-Ivic-Matula（GIM）函数g（n）计算Matula数为n的唯一树的边数。由于有一个前缀免费的代码的一组素数，使得每个素数p的codelength是2g（p），我们展示了如何一些性质的GIM功能可以得到平凡的编码定理。
Denote by $p_m$ the $m$-th prime number ($p_1=2,~p_2=3,~p_3=5,~ p_4=7,~\ldots$). Let $T$ be a rooted tree with branches $T_1,T_2,\ldots,T_r$. The Matula number $M(T)$ of $T$ is $p_{M(T_1)}\cdot p_{M(T_2)}\cdot \ldots \cdot p_{M(T_r)}$, starting with $M(K_1)=1$. This number was put forward half a century ago by the American mathematician David Matula. In this paper, we prove that the star (consisting of a root and leaves attached to it) and the binary caterpillar (a binary tree whose internal vertices form a path starting at the root) have the smallest and greatest Matula number, respectively, …
With graphs considered as natural models for many network design problems, edge connectivity $κ'(G)$ and maximum number of edge-disjoint spanning trees $τ(G)$ of a graph $G$ have been used as measures for reliability and strength in communication networks modeled as graph $G$ (see \cite{Cunn85, Matula87}, among others). Mader \cite{Mader71} and Matula \cite{Matula72} introduced the maximum subgraph edge connectivity $\overline{κ'}(G)=\max \{κ'(H): H \mbox{ is a subgraph of } G \}$. Motivated by their applications in network design and by the established inequalities \[ \overline{κ'}(G)\ge κ'(G…
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We develop algorithms, implemented in Maple, that study the number of vertices with a particular number of children in a random ordered tree where all vertices must have a number of children in some finite set. By calculating the mixed moments of two such numbers, the package gives strong experimental evidence numbers are pairwise asymptotically normal.
Decompositions and statistics for beta(1,0)-trees and nonseparable permutations
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k-non-crossing trees and edge statistics modulo k
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On Weary Drivers, Records of Trees, and Parking Functions
本文基于根树的记录概念，重新定义停车函数，并建立根树与停车函数之间的记录保持双射，同时证明两者统计量的等分布性，并探讨了若干计数问题。
A rooted tree T is a poset whose Hasse diagram is a graph-theoretic tree having a unique minimal element. We study rowmotion on antichains and lower order ideals of T. Recently Elizalde, Roby, Plante and Sagan considered rowmotion on fences which are posets whose Hasse diagram is a path (but permitting any number of minimal elements). They showed that in this case, the orbits could be described in terms of tilings of a cylinder. They also defined a new notion called homometry which means that a statistic takes a constant value on all orbits of the same size. This is a weaker condition than the…
A branch statistic for trees: Interpreting coefficients of the characteristic polynomial of braid deformations
本文针对辫变形超平面配置，定义了标号树上的分支统计量，使其分布由特征多项式系数给出，适用于 extended Catalan、Shi、Linial 和 semiorder 等族。
Exploring Werner Krandick's Binary Tree Jump Statistics
本文利用实验数学和符号计算推导了二叉树跳跃统计量的闭式表达式及高阶矩，并探讨其渐近正态性，同时引用了关于渐近正态性的完整证明。
A hyperbinary expansion of a positive integer n is a partition of n into powers of 2 in which each part appears at most twice. In this paper, we consider a generalization of this concept and a certain statistic on the corresponding set of expansions of n. We then define q-generalized m-ary trees whose vertices are labeled by ratios of two consecutive terms within the sequence of distribution polynomials for the aforementioned statistic. When m = 2, we obtain a variant of a previously considered q-Calkin-Wilf tree.
Two involutions on binary trees and generalizations
本文研究了二叉树上的两个对合及其推广，结合经典双射$φ$解决了多个公开的双射问题，并导出了有根树上的新统计量与四元等分布。
Symmetry in Tree Parking Distributions
本文利用首次回归分解研究毛虫树停车分布中幸运车辆数与车位偏好频率的联合对称性，构造并证明 Fuss-Catalan 生成函数的 $q,t$-模拟，并推广到任意 $m$ 个统计量。
An involution on β(1,0)-trees
在[Decompositions and statistics for β（1，0）-trees and nonseparable permutations，Advances Appl. Math.42（2009）313--328]中，我们在β（1，0）-trees上引入了对合h。然而，我们忽略了证明h确实是一个对合。在本说明中，我们提供了缺失的证据。我们还改进了在同一篇文章中给出的一个等分布结果。
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研究随机图中最大树扩展计数的渐进行为，揭示不同稀疏度下M_n的行为变化机制。
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Sparse Random Matrices and Statistics of Rooted Trees
我们考虑平均每行具有p个非零元素的N维随机对称矩阵A的系综。本文研究了A的范数在N和p无限递增的极限下的渐近性态，证明了p= logN是范数有界或无界的临界值.参数是基于树型图的演算。稀疏随机矩阵的渐近性质基本上取决于树的顶点，我们证明是有限的典型程度。
Statistics of planar graphs viewed from a vertex: A study via labeled trees
我们研究随机平面四边形剖分和欧拉三角剖分中参考顶点（原点）附近的边和顶点的统计。对于具有固定边数和顶点数且距原点测地距离给定的图，得到了精确的生成函数。我们的分析依赖于双射与标签树，其中的标签编码的信息上的测地线距离的原点。在无限大的图形的情况下，我们给出了特别明确的公式的概率，原点已给定的数量的相邻的边缘和/或顶点，以及明确的值为相应的时刻。
Large fringe trees for random trees with given vertex degrees
本文研究了具有给定顶点度数的随机平面树中边树的统计性质，探讨了目标树随随机树规模增长时的计数问题，采用四种概率方法分析了子树计数的泊松和正态收敛条件。
An algorithm for uniform generation of unlabeled trees (Pólya trees), with an extension of Cayley's formula
本文提出高效算法均匀生成波利亚树，能对比无标签与有标签树统计特性及渐近定理与实际情况，并给出给定自同构下保持不变的有根有标签树数量的乘积公式，细化了凯莱公式。
Distance statistics in large toroidal maps
我们计算了一些距离依赖的通用尺度函数，其特征在于亏格1的大型地图的距离统计。特别地，我们得到明确的概率分布的最短的不可收缩回路通过一个随机点的地图，并为两个随机点之间的距离的长度。我们的研究结果是在二分环形四边形的背景下，使用他们的编码以及标记的1-树，这是一个单一的脸和适当的整数顶点标签的属一地图。在这个框架内，上面的分布可以简单地作为良好标记的1-树的适当生成函数的缩放极限来获得，所有这些都可以用少量良好标记的平面树的基本缩放函数来表示。
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