On the poset of vector partitions

TL;DR

This paper studies the poset of vector partitions, extending known shellability results to edge-lexicographic shellability, and relates the number of spheres in the homotopy type to combinatorial objects like non-ambiguous trees.

Contribution

It proves that the poset of vector partitions is edge-lexicographically shellable and provides a recursive formula for counting the spheres, linking to combinatorial structures.

Findings

Poset of vector partitions is edge-lexicographically shellable.

Recursive formula for the number of spheres in the homotopy type.

Number of spheres when s=1 equals the count of non-ambiguous trees.

Abstract

We consider the poset of vector partitions of $[n]$ into $s$ components, denoted $\Pi_{n,s}$, which was first defined by Stanley in 1978. In 1986, Sagan showed that this poset is CL-shellable, and hence has the homotopy type of a wedge of spheres of dimension $(n-2)$. We extend on this result to show that $\Pi_{n,s}$ is edge-lexicographic shellable. We then use this edge-labeling to 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, first defined in 2014 by Aval, Boussicault, Bouvel and Silimbani.