Counting Line-Colored D-ary 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 $\frac{1}{{\sum }_{i=1}^D{p}_i+1}\left(\genfrac{}{}{0ex}{}{{\sum }_{i=1}^D{p}_i+1}{p_1}\right)...\left(\genfrac{}{}{0ex}{}{{\sum }_{i=1}^D{p}_i+1}{p_D}\right)$.

6 pages

References in corpus (1)

• Random tensor models in the large N limit: Uncoloring the colored tensor models