Trees and cycles

TL;DR

This paper investigates how different trees on n vertices generate cyclic permutations through transpositions, analyzing the number of cycles realized, their distributions, and related combinatorial sequences.

Contribution

It determines the number of realized cycles for various trees and solves the inverse problem of identifying trees that produce a given cycle.

Findings

Number of realized cycles for different trees

Distribution of realizations of each cycle

Connections to Euler and Fuss–Catalan numbers

Abstract

Let $T$ be a tree on $n$ vertices. We can regard the edges of $T$ as transpositions of the vertex set; their product (in any order) is a cyclic permutation. All possible cyclic permutations arise (each exactly once) if and only if the tree is a star. In this paper we find the number of realised cycles, and obtain some results on the number of realisations of each cycle, for other trees. We also solve the inverse problem of the number of trees which give rise to a given cycle. On the way, we meet some familiar number sequences including the Euler and Fuss--Catalan numbers.