Combinatorica

, Volume 12, Issue 3, pp 247–260

Counting colorful multi-dimensional trees

Authors

  • Ron M. Adin
    • Institute of MathematicsHebrew University
Article

DOI: 10.1007/BF01285814

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

Abstract

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 5005 C 0505 C 3005 C 6515 A 18

Copyright information

© Akademiai Kiado 1992