Article

Israel Journal of Mathematics

, Volume 45, Issue 4, pp 337-351

Enumeration ofQ-acyclic simplicial complexes

  • Gil KalaiAffiliated withInstitute of Mathematics, The Hebrew University of Jerusalem

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Abstract

Let
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(n, k) be the class of all simplicial complexesC over a fixed set ofn vertices (2≦k≦n) such that: (1)C has a complete (k−1)-skeleton, (2)C has precisely ( k n−1 )k-faces, (3)H k (C)=0. We prove that for
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,H k−1(C) is a finite group, and our main result is:
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. This formula extends to high dimensions Cayley’s formula for the number of trees onn labelled vertices. Its proof is based on a generalization of the matrix tree theorem.