Hyperarbres, arbres enracin\'{e}s et partitions point\'{e}es

Abstract

We compute the characteristic polynomials of the posets of hypertrees. We show that the generating series of the polynomials can be expressed using cyclic hypertrees. We also propose a conjecture on the action of the symmetric groups on the homology of these posets. On the other hand, we show that Vallette's poset of pointed partitions is homotopy equivalent to Pitman's poset of forests. The implicit common thema of the article is the combinatorics of the PreLie operad.