Decomposition of Triply Rooted Trees

TL;DR

This paper introduces a novel decomposition of triply rooted trees into three doubly rooted trees, providing a combinatorial interpretation of a conjectured identity in machine learning theory and establishing a bijection with certain functions.

Contribution

It presents a new combinatorial decomposition and a bijection that connect triply rooted trees with functions, advancing understanding in combinatorics and machine learning theory.

Findings

Decomposition of triply rooted trees into three doubly rooted trees.

A combinatorial interpretation of Lacasse's identity.

A bijection between functions from [n+1] to [n] and triply rooted trees.

Abstract

In this paper, we give a decomposition of triply rooted trees into three doubly rooted trees. This leads to a combinatorial interpretation of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory, and proved by Younsi by using the Hurwitz identity on multivariate Abel polynomials. We also give a bijection between the set of functions from $[n+1]$ to $[n]$ and the set of triply rooted trees on $[n]$, which leads to the refined enumeration of functions from $[n+1]$ to $[n]$ with respect to the number of elements in the orbit of $n+1$ and the number of periodic points.