A bijective enumeration of labeled trees with given indegree sequence

TL;DR

This paper introduces a bijective method to enumerate labeled trees with a specified indegree sequence, proving a conjecture and solving open problems, while also establishing a related binomial coefficient identity.

Contribution

It constructs a bijection linking unrooted and rooted trees with matching indegree sequences, providing a combinatorial proof of a recent conjecture and solving open problems.

Findings

Established a bijection between unrooted and rooted trees with matching indegree sequences.

Provided a bijective proof confirming Cotterill's conjecture.

Proved a q-multisum binomial coefficient identity related to the problem.

Abstract

For a labeled tree on the vertex set $\set{1,2,\ldots,n}$, the local direction of each edge $(i\,j)$ is from $i$ to $j$ if $i<j$. For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges pointing to a vertex is called its indegree. Thus the local (resp. global) indegree sequence $\lambda = 1^{e_1}2^{e_2} \ldots$ of a tree on the vertex set $\set{1,2,\ldots,n}$ is a partition of $n-1$. We construct a bijection from (unrooted) trees to rooted trees such that the local indegree sequence of a (unrooted) tree equals the global indegree sequence of the corresponding rooted tree. Combining with a Pr\"ufer-like code for rooted labeled trees, we obtain a bijective proof of a recent conjecture by Cotterill and also solve two open problems proposed by Du and Yin. We also prove a $q$-multisum binomial coefficient identity which confirms another conjecture of Cotterill in a very special case.