Semi-pointed partition posets and Species

TL;DR

This paper introduces semi-pointed partition posets, generalizes partition posets, proves their Cohen-Macaulay property, and explores their algebraic structures and symmetric group actions on homology.

Contribution

It defines semi-pointed partition posets, establishes their Cohen-Macaulay property, and analyzes their algebraic and symmetric group action structures.

Findings

Semi-pointed partition posets are Cohen-Macaulay.

The dimension and character of symmetric group actions are computed.

The associated incidence Hopf algebra resembles the Faà di Bruno Hopf algebra.

Abstract

We define semi-pointed partition posets, which are a generalisation of partition posets and show that they are Cohen-Macaulay. We then use multichains to compute the dimension and the character for the action of the symmetric groups on their homology. We finally study the associated incidence Hopf algebra, which is similar to the Fa{\`a} di Bruno Hopf algebra.