scholarly journals On an Identity for the Cycle Indices of Rooted Tree Automorphism Groups

10.37236/1152 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Stephan G. Wagner
Keyword(s):  
Symmetric Group ◽  
Elementary Proof ◽  
Automorphism Groups ◽  
Rooted Tree ◽  
Rooted Trees ◽  
Cycle Index ◽  
Spanning Forest ◽  
Cycle Indices

This note deals with a formula due to G. Labelle for the summed cycle indices of all rooted trees, which resembles the well-known formula for the cycle index of the symmetric group in some way. An elementary proof is provided as well as some immediate corollaries and applications, in particular a new application to the enumeration of $k$-decomposable trees. A tree is called $k$-decomposable in this context if it has a spanning forest whose components are all of size $k$.

CONJUGATION IN TREE AUTOMORPHISM GROUPS

2001 ◽  
Vol 11 (05) ◽  
pp. 529-547 ◽  
Author(s):  
PIOTR W. GAWRON ◽  
VOLODYMYR V. NEKRASHEVYCH ◽  
VITALY I. SUSHCHANSKY
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Rooted Tree ◽  
Symmetric Groups ◽  
Wreath Products ◽  
Rooted Trees ◽  
Full Description ◽  
Finite Tree ◽  
Locally Finite ◽  
Infinite Sequences

It is given a full description of conjugacy classes in the automorphism group of the locally finite tree and of a rooted tree. They are characterized by their types (a labeled rooted trees) similar to the cyclical types of permutations. We discuss separately the case of a level homogenous tree, i.e. conjugality in wreath products of infinite sequences of symmetric groups. It is proved those automorphism groups of rooted and homogenous non-rooted trees are ambivalent.

scholarly journals Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group

2017 ◽  
Vol 5 (1) ◽  
pp. 158-201
Author(s):  
Jan Brandts ◽  
Apo Cihangir
Keyword(s):  
Convex Hull ◽  
Computer Program ◽  
Explicit Form ◽  
Linear Algebra ◽  
Dihedral Angles ◽  
Cycle Index ◽  
Hyperoctahedral Group ◽  
Diagonally Dominant ◽  
Group B ◽  
Cycle Indices

Abstract The convex hull of n + 1 affinely independent vertices of the unit n-cube In is called a 0/1-simplex. It is nonobtuse if none its dihedral angles is obtuse, and acute if additionally none of them is right. In terms of linear algebra, acute 0/1-simplices in In can be described by nonsingular 0/1-matrices P of size n × n whose Gramians G = PTP have an inverse that is strictly diagonally dominant, with negative off-diagonal entries [6, 7]. The first part of this paper deals with giving a detailed description of how to efficiently compute, by means of a computer program, a representative from each orbit of an acute 0/1-simplex under the action of the hyperoctahedral group Bn [17] of symmetries of In. A side product of the investigations is a simple code that computes the cycle index of Bn, which can in explicit form only be found in the literature [11] for n ≤ 6. Using the computed cycle indices for B3, . . . ,B11 in combination with Pólya’s theory of enumeration shows that acute 0/1-simplices are extremely rare among all 0/1-simplices. In the second part of the paper, we study the 0/1-matrices that represent the acute 0/1-simplices that were generated by our code from a mathematical perspective. One of the patterns observed in the data involves unreduced upper Hessenberg 0/1-matrices of size n × n, block-partitioned according to certain integer compositions of n. These patterns will be fully explained using a so-called One Neighbor Theorem [4]. Additionally, we are able to prove that the volumes of the corresponding acute simplices are in one-to-one correspondence with the part of Kepler’s Tree of Fractions [1, 24] that enumerates ℚ ⋂ (0, 1). Another key ingredient in the proofs is the fact that the Gramians of the unreduced upper Hessenberg matrices involved are strictly ultrametric [14, 26] matrices.

Cyclic Renormalization and Automorphism Groups of Rooted Trees

1996 ◽  
Author(s):  
Hyman Bass ◽  
Maria Victoria Otero-Espinar ◽  
Daniel Rockmore ◽  
Charles Tresser
Keyword(s):  
Automorphism Groups ◽  
Rooted Trees

Automorphism groups of rooted trees have property (FA): A new proof

2012 ◽  
Vol 15 (2) ◽  
Author(s):  
Maciej Malicki
Keyword(s):  
Automorphism Groups ◽  
Rooted Trees

Abstract.We give a new proof of a theorem of Psaltis

scholarly journals On Sackin's original proposal: The variance of the leaves' depths as a phylogenetic balance index

2019 ◽  
Author(s):  
Tomás Martínez Coronado ◽  
Arnau Mir ◽  
Francesc Rossello ◽  
Lucía Rotger
Keyword(s):  
Phylogenetic Tree ◽  
Phylogenetic Trees ◽  
Rooted Tree ◽  
Rooted Trees ◽  
Closed Formula ◽  
Original Proposal ◽  
Yule Model ◽  
Uniform Model ◽  
Rooted Phylogenetic Tree ◽  
Balance Index

Abstract Background: The Sackin index S of a rooted phylogenetic tree, defined as the sum of its leaves' depths, is one of the most popular balance indices in phylogenetics, and Sackin's 1972 paper is usually cited as the source for this index. However, what Sackin actually proposed in his paper as a measure of the imbalance of a rooted tree was not the sum of its leaves' depths, but their "variation". This proposal was later implemented as the variance of the leaves' depths by Kirkpatrick and Slatkin, where moreover they posed the problem of finding a closed formula for its expected value under the Yule model. Nowadays, Sackin's original proposal seems to have passed into oblivion in the phylogenetics literature, replaced by the index bearing his name, which, in fact, was introduced a decade later by Sokal.Results: In this paper we study the properties of the variance of the leaves' depths, V, as a balance index. Firstly, we prove that the rooted trees with n leaves and maximum V value are exactly the combs with n leaves. But although V achieves its minimum value on every space BT_n of bifurcating rooted phylogenetic trees with n< 184 leaves at the so-called "maximally balanced trees" with n leaves, this property fails for almost every n>= 184. We provide then an algorithm that finds in O(n) time the trees in BT_n with minimum V value. Secondly, we obtain closed formulas for the expected V value of a bifurcating rooted tree with any number n of leaves under the Yule and the uniform models and, as a by-product of the computations leading to these formulas, we also obtain closed formulas for the variance of the Sackin index and the total cophenetic indexof a bifurcating rooted tree, as well as of their covariance, under the uniform model, thus filling this gap in the literature.Conclusions: The phylogenetics crowd has been wise in preferring as a balance index the sum S(T) of the leaves’ depths of a phylogenetic tree T over their variance V (T), because the latter does not seem to capture correctly the notion of balance of large bifurcating rooted trees. But for bifurcating trees up to 183 leaves, V is a valid and useful balance index.

scholarly journals A Refinement of Cayley's Formula for Trees

10.37236/1884 ◽  
2006 ◽  
Vol 11 (2) ◽  
Author(s):  
Ira M. Gessel ◽  
Seunghyun Seo
Keyword(s):  
Rooted Tree ◽  
Rooted Trees ◽  
Binary Trees ◽  
Parking Functions ◽  
Hook Length ◽  
Proper Vertex

A proper vertex of a rooted tree with totally ordered vertices is a vertex that is the smallest of all its descendants. We count several kinds of labeled rooted trees and forests by the number of proper vertices. Our results are all expressed in terms of the polynomials $$P_n(a,b,c)= c\prod_{i=1}^{n-1}(ia+(n-i)b +c),$$ which reduce to $(n+1)^{n-1}$ for $a=b=c=1$. Our study of proper vertices was motivated by Postnikov's hook length formula $$(n+1)^{n-1}={n!\over 2^n}\sum _T \prod_{v}\left(1+{1\over h(v)}\right),$$ where the sum is over all unlabeled binary trees $T$ on $n$ vertices, the product is over all vertices $v$ of $T$, and $h(v)$ is the number of descendants of $v$ (including $v$). Our results give analogues of Postnikov's formula for other types of trees, and we also find an interpretation of the polynomials $P_n(a,b,c)$ in terms of parking functions.

Realization of plucking polynomials

2017 ◽  
Vol 26 (02) ◽  
pp. 1740016 ◽  
Author(s):  
Zhiyun Cheng ◽  
Sujoy Mukherjee ◽  
Józef H. Przytycki ◽  
Xiao Wang ◽  
Seung Yeop Yang
Keyword(s):  
Rooted Tree ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Rooted Trees ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for a given polynomial to be a plucking polynomial of a rooted tree. We discuss the fact that different rooted trees can have the same polynomial.

scholarly journals The enumeration of rooted trees by total height

1969 ◽  
Vol 10 (3-4) ◽  
pp. 278-282 ◽  
Author(s):  
John Riordan ◽  
N. J. A. Sloane
Keyword(s):  
Neural Networks ◽  
Rooted Tree ◽  
Rooted Trees ◽  
Total Height ◽  
Random Neural Networks

The height (as in [3] and [4]) of a point in a rooted tree is the length of the path (that is, the number of lines in the path) from it to the root; the total height of a rooted tree is the sum of the heights of its points. The latter arises naturally in studies of random neural networks made by one of us (N.J.A.S.), where the enumeration of greatest interest is that of trees with all points distinctly labeled.

Sign in / Sign up

Export Citation Format

Share Document