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### Horizontal visibility graph of a random restricted growth sequence.

##### Authors:
Toufik Mansour Reza Rastegar Alexander Roitershtein

Adv Appl Math 2021 Mar 9;124. Epub 2020 Dec 9.

Department of Statistics, Texas A&M University, College Station, TX 77843, USA.

We study the distributional properties of horizontal visibility graphs associated with random restrictive growth sequences and random set partitions of size . Our main results are formulas expressing the expected degree of graph nodes in terms of simple explicit functions of a finite collection of Stirling and Bernoulli numbers. Read More

### Roadblocked monotonic paths and the enumeration of coalescent histories for non-matching caterpillar gene trees and species trees.

##### Authors:
Zoe M Himwich Noah A Rosenberg

Adv Appl Math 2020 Feb 31;113. Epub 2019 Oct 31.

Department of Biology, Stanford University, Stanford, CA 94305 USA.

Given a gene tree topology and a species tree topology, a coalescent history represents a possible mapping of the list of gene tree coalescences to associated branches of a species tree on which those coalescences take place. Enumerative properties of coalescent histories have been of interest in the analysis of relationships between gene trees and species trees. The simplest enumerative result identifies a bijection between coalescent histories for a matching caterpillar gene tree and species tree with monotonic paths that do not cross the diagonal of a square lattice, establishing that the associated number of coalescent histories for -taxon matching caterpillar trees ( ⩾ 2) is the Catalan number . Read More

February 2020

### ENUMERATION OF LONELY PAIRS OF GENE TREES AND SPECIES TREES BY MEANS OF ANTIPODAL CHERRIES.

##### Authors:
Noah A Rosenberg

Adv Appl Math 2019 Jan 14;102:1-17. Epub 2018 Sep 14.

Department of Biology, Stanford University, Stanford, CA 94305 USA.

In mathematical phylogenetics, given a rooted binary leaf-labeled gene tree topology and a rooted binary leaf-labeled species tree topology with the same leaf labels, a coalescent history represents a possible mapping of the list of gene tree coalescences to the associated branches of the species tree on which those coalescences take place. For certain families of ordered pairs (), the number of coalescent histories increases exponentially or even faster than exponentially with the number of leaves . Other pairs have only a single coalescent history. Read More

January 2019

### RECOVERING A TREE FROM THE LENGTHS OF SUBTREES SPANNED BY A RANDOMLY CHOSEN SEQUENCE OF LEAVES.

##### Authors:
Steven N Evans Daniel Lanoue

Adv Appl Math 2018 May 28;96:39-75. Epub 2018 Feb 28.

Department of Mathematics, University of California, 970 Evans Hall #3840, Berkeley, CA 94720-3840, U.S.A.

Given an edge-weighted tree with leaves, sample the leaves uniformly at random without replacement and let , 2 ≤ ≤ , be the length of the subtree spanned by the first leaves. We consider the question, "Can be identified (up to isomorphism) by the joint probability distribution of the random vector (, …, )?" We show that if is known to belong to one of various families of edge-weighted trees, then the answer is, "Yes." These families include the edge-weighted trees with edge-weights in general position, the ultrametric edge-weighted trees, and certain families with equal weights on all edges such as ( + 1)-valent and rooted -ary trees for ≥ 2 and caterpillars. Read More