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arXiv · 2503.22110

Equivalences and Distinctions in Lexicographic Shellability of Posets

Stephen Lacina·Grace Stadnyk
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Abstract

We present two perhaps surprisingly small posets, one graded and one non-graded, that are CC-shellable in the sense of Kozlov and TCL-shellable in the sense of Hersh, but not CL-shellable in the sense of Björner and Wachs. In the spirit of Björner and Wachs' recursive atom orderings (RAO) and Hersh and Stadnyk's generalized recursive atom orderings (GRAO), we also introduce a notion called recursive first atom sets (RFAS). An RFAS is a set of conditions on the atoms of each interval in a finite bounded poset $P$ that are necessary for CC-shellability of $P$ and sufficient for shellability of $P$. We also prove that under an extra condition, $P$ is CC-shellable if and only if it admits an RFAS, in the same way that RAOs provide a reformulation of CL-shellability.

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Stephen Lacina, Grace Stadnyk. 2025-03-28. Equivalences and Distinctions in Lexicographic Shellability of Posets. https://arxiv.org/abs/2503.22110

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