The poset of $s$-labeled partitions of $[n]$

Natalie Aisbett

Published 2015-05-08Version 1

We define a poset called the poset of $s$-labeled partitions of the $[n]$, denoted $\Pi_{n}^s$, where $s$ is an integer greater than 0. We show that the order complex of $\overline{\Pi}_{n}^s$ has the homotopy type of a wedge of spheres of dimension $(n-2)$ by showing that $\Pi_{n}^s$ is edge-lexicographic shellable. We then find a recursive expression for the number of spheres, and show that when $s=1$ the number of spheres is equal to the number of complete non-ambiguous trees.