Combinatorica

, Volume 12, Issue 3, pp 247–260

Counting colorful multi-dimensional trees

  • Ron M. Adin
Article

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 50 05 C 05 05 C 30 05 C 65 15 A 18 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    T. L. Austin: The enumeration of point labelled chromatic graphs and trees,Canad. J. of Math. 12 (1960), 535–545.Google Scholar
  2. [2]
    E. D. Bolker: Simplicial geometry and transportation polytopes,Trans. AMS 217 (1976), 121–142.Google Scholar
  3. [3]
    A. Cayley: A theorem on trees,Quarterly J. of Math. 23 (1889), 376–378.Google Scholar
  4. [4]
    M. Fiedler andJ. Sedláček: O W-basich orientovaných grafu,Časopis pro pěstováni matematiky 83 (1958), 214–225.Google Scholar
  5. [5]
    F. R. Gantmacher:The Theory of Matrices, Chelsea, 1960.Google Scholar
  6. [6]
    G. Kalai: Enumeration ofQ-acyclic simplicial complexes,Israel J. of Math. 45 (1983), 337–351.Google Scholar
  7. [7]
    J. W. Moon: Counting Labelled Trees,Canadian Mathematical Congress, (1970).Google Scholar
  8. [8]
    J. R. Munkres:Elements of Algebraic Topology, Benjamin/Cummings, 1984.Google Scholar
  9. [9]
    E. H. Spanier:Algebraic Topology, McGraw-Hill, 1966.Google Scholar

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

Personalised recommendations