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arXiv · 2606.08724

EL-Shellability of the poset of ranked cactuses

Vincent Moulton·Andreas Spillner·Antonia Stavemann
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Abstract

Recently the poset of ranked cactuses $(\mathfrak{P}(X),\preceq)$ was introduced. For a finite set $X$, this poset consists of a set $\mathfrak{P}(X)$ of certain collections of ordered pairs of subsets of $X$ together with an ordering $\preceq$ that is similar to the refinement ordering of partitions of a finite set. In addition, the maximal chains in this poset correspond to binary ranked cactuses, a fact which can be used to construct the so-called space of equidistant cactuses. In this paper, we show that the poset of ranked cactuses is EL-shellable. As a consequence we also show that the proper part of the link of the origin of the space of equidistant cactuses has the homotopy type of a wedge of spheres.

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Vincent Moulton, Andreas Spillner, Antonia Stavemann. 2026-06-07. EL-Shellability of the poset of ranked cactuses. https://arxiv.org/abs/2606.08724

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