, Volume 12, Issue 3, pp 247–260

Counting colorful multi-dimensional trees

  • Ron M. Adin

DOI: 10.1007/BF01285814

Cite this article as:
Adin, R.M. Combinatorica (1992) 12: 247. doi:10.1007/BF01285814


LetV be a disjoint union ofr finite setsV1,...,Vr (“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 ΔT

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.

AMS subject classification code (1991)

05 C 50 05 C 05 05 C 30 05 C 65 15 A 18 

Copyright information

© Akademiai Kiado 1992

Authors and Affiliations

  • Ron M. Adin
    • 1
  1. 1.Institute of MathematicsHebrew UniversityJerusalemIsrael
  2. 2.Department of MathematicsBar-Han UniversityRamat-GabIsrael

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