## Abstract

Let V be a disjoint union of r finite sets V_{1},..., V_{r} ("colors"). A collection T of subsets of V is colorful if each member if T contains at most one point of each color. A k-dimensional colorful tree is a colorful collection T of subsets of V, each of size k+1, such that if we add to T all the colorful subsets of V of size k or less, we get a Q-acyclic simplicial complex Δ_{T} We count (using the Binet-Cauchy theorem) the k-dimensional colorful trees on V (for all k), where each tree T is counted with weight {Mathematical expression}. The result confirms, in a way, a formula suggested by Bolker. (for k-r-1). It extends, on one hand, a result of Kalai on weighted counting of k-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.

Original language | English |
---|---|

Pages (from-to) | 247-260 |

Number of pages | 14 |

Journal | Combinatorica |

Volume | 12 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1992 |

Externally published | Yes |

## Keywords

- AMS subject classification code (1991): 05C50, 05C05, 05C30, 05C65, 15A18