Abstract
We present here a family of posets which generalizes both partition and pointed partition posets. After a short description of these new posets, we show that they are Cohen-Macaulay, compute their Moebius numbers and their characteristic polynomials. The characteristic polynomials are obtained using a combinatorial interpretation of the incidence Hopf algebra associated to these posets. Nous introduisons ici une famille de posets qui généralise à la fois les poset de partitions et les posets de partitions pointées. Après une description rapide de ces nouveaux posets, nous montrons qu’ils sont Cohen-Macaulay et nous calculons leurs nombres de Moebius et leurs polynômes caractéristiques. Ces derniers sont obtenus grâce à une interprétation combinatoire de l’algèbre de Hopf d’incidence associée à ces posets.
Highlights
The partition poset on a finite set V is the well-known poset of partitions of V, endowed with the following partial order: a partition P is smaller than another partition Q if the parts of Q are unions of parts of P
The pointed partition poset on V is the set of pointed partitions of V, where a pointed partition P is smaller than another pointed partition Q if and only if the parts of Q are unions of parts of P and the set of pointed elements of Q is contained in the set of pointed elements of P
After a short description of semi-pointed partition posets, we show that these posets are Cohen-Macaulay thanks to total semi-modularity
Summary
Example 1.2 There are 35 (4, 2)-semi-pointed partitions. Remark 1.3 A structure of 2-coloured operad is hidden in the decoration of partitions described in the definition of semi-pointed partitions. The set of semi-pointed partitions on V = V1 V2 can be endowed with the following partial order: Definition 1.4 Let P and Q be two semi-pointed partitions. We denote by ΠV,V1 the poset of semi-pointed partitions of V = V1 V2 bounded by the addition of a greatest element 1 and Πn, the poset of (n, )-semi-pointed partitions bounded by the addition of a greatest element 1. The least element of the poset ΠV,V1 , which is the partition whose parts are of cardinality 1, endowed with the only possible pointing, will be denoted by πV,V1. It comes by identifying non pointed parts with parts pointed in the last element n
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
