Pointed and multi-pointed partitions of type A and B

Abstract

21 pages, 3 figures.International audienceThe aim of this paper is to define and study pointed and multi-pointed partition posets of type A and B (in the classification of Coxeter groups). We compute their characteristic polynomials, incidence Hopf algebras and homology groups. As a corollary, we get that some operads are Koszul over Z. Cet article introduit et etudie les posets des partitions pointees et multi-pointees en type A et B (dans la classification des groupes de Coxeter finis). Leurs polynomes caracteristiques, leurs algebres de Hopf d'incidence et leurs groupes d'homologie sont calcules. Comme corollaire, on obtient que certaines operades sont de Koszul sur Z

Homology of generalized partition posets

“…We refer the reader to the article [6] of Chapoton and the author for a study of the properties of the pointed partition posets.…”

“…In [6], we have shown that the posets of multi-pointed partitions are totally semi-modular. Therefore, they are Cohen-Macaulay and we have the following theorem.…”

“…In [6], we prove that the intervals of maximal length in the posets of multi-pointed partitions are totally semimodular. It shows that they are Cohen-Macaulay over any field k and over the ring of integers Z.…”

“…It is worth mentioning here that in [6], Chapoton and the author studied the proprieties of the pointed and multipointed posets and applied the results of this paper to prove that the operads Perm and ComT rias are Koszul over Z.…”

Hyperarbres, arbres eracinés et partitions pointés

“…On peut obtenir une preuve plus courte de cette proposition en utilisant les résultats de [7] et les méthodes de [20] qui donnent un description du complexe d'ordre de chaque intervalle dans le poset des partitions pointées.…”