The Cactus Group Property for Ordinal Sums of Disjoint Unions of Chains

TL;DR

This paper characterizes when ordinal sums of disjoint unions of chains satisfy cactus relations under Bender-Knuth involutions, extending the understanding of LE-cactus posets beyond d-complete cases.

Contribution

It generalizes the class of LE-cactus posets to include ordinal sums of disjoint unions of chains, providing a broader characterization.

Findings

Ordinal sums of disjoint unions of chains can be LE-cactus under certain conditions.

The paper extends the class of known LE-cactus posets beyond d-complete posets.

It offers a new framework for understanding cactus relations in poset linear extensions.

Abstract

We study the action of Bender-Knuth involutions on linear extensions of posets and identify LE-cactus posets, i.e. those for which the cactus relations hold. It was conjectured in \cite{chiang2023bender} that d-complete posets are LE-cactus. Among the non-d-complete posets that are LE-cactus, one notable family is ordinal sums of antichains. In this paper, we characterize the LE-cactus posets in a more general family, namely ordinal sums of disjoint unions of chains.