Counting Line-Colored D-ary Trees

TL;DR

This paper studies a generating function for counting specific colored rooted trees related to tensor models, providing an explicit formula for the number of such trees with given color distributions.

Contribution

It introduces an independent method to derive a closed-form formula for counting line-colored D-ary trees with specified edge counts per color.

Findings

Derived an explicit formula for the number of line-colored D-ary trees with given color counts.

Connected the combinatorial enumeration to observables in tensor models.

Provided a new proof technique for counting these specialized trees.

Abstract

Random tensor models are generalizations of matrix models which also support a 1/N expansion. The dominant observables are in correspondence with some trees, namely rooted trees with vertices of degree at most $D$ and lines colored by a number $i$ from 1 to $D$ such that no two lines connecting a vertex to its descendants have the same color. In this Letter we study by independent methods a generating function for these observables. We prove that the number of such trees with exactly $p_i$ lines of color $i$ is $\frac{1}{\sum_{i=1}^D p_i +1} \binom{\sum_{i=1}^D p_i+1}{p_1} ... \binom{\sum_{i=1}^D p_i+1}{p_D}$.