, Volume 12, Issue 3, pp 247-260

Counting colorful multi-dimensional trees

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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.