Counting colorful multi-dimensional trees

Abstract

LetV be a disjoint union ofr finite setsV ₁,...,V _r (“colors”). A collectionT of subsets ofV iscolorful if each member ifT contains at most one point of each color. Ak-dimensional colorful tree is a colorful collectionT of subsets ofV, each of sizek+1, such that if we add toT all the colorful subsets ofV of sizek or less, we get aQ-acyclic simplicial complex Δ _T

We count (using the Binet-Cauchy theorem) thek-dimensional colorful trees onV (for allk), where each treeT is counted with weight$|\tilde H_{k - 1} (\Delta _T )|^2 (\tilde H_* = reduced homology)$. The result confirms, in a way, a formula suggested by Bolker. (fork-r−1). It extends, on one hand, a result of Kalai on weighted counting ofk-dimensional trees and, on the other hand, enumeration formulas for multi-partite (1-dimensional) trees. All these results are extensions of Cayley's celebrated treecounting formula, now 100 years old.