Poset Partitions and the Combinatorics of the $\textbf{cd}$-Index

TL;DR

This paper introduces S-partitionable and SE-partitionable posets, new classes of Eulerian and semi-Eulerian posets, demonstrating their non-negativity of the cd-index through recursive formulas and generalizing previous shellability concepts.

Contribution

It defines S-partitionable and SE-partitionable posets and proves their non-negative cd-index, extending the combinatorial understanding of Eulerian posets.

Findings

S-partitionable posets have a non-negative cd-index.

SE-partitionable posets also have a non-negative semi-Eulerian cd-index.

The paper generalizes S-shellable complexes and semi-Eulerian posets.

Abstract

We introduce a new class of Eulerian posets, called S-partitionable posets, which have a non-negative cd-index. These posets are a generalization of S-shellable complexes introduced by Stanley in 1994. We prove that S-partitionable posets have a non-negative cd-index via a recursive formula. Then, we introduce a semi-Eulerian version of S-partitionable posets, which we call SE-partitionable posets. We show that SE-partitionable posets also have a non-negative semi-Eulerian cd-index as defined by Juhnke-Kubitzke, Samper and Venturello in 2024.