Generalized recursive atom ordering and equivalence to CL-shellability
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913476025253888 |
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| author | Hersh, Patricia Stadnyk, Grace |
| author_facet | Hersh, Patricia Stadnyk, Grace |
| contents | Björner and Wachs introduced CL-shellability as a technique for studying the topological structure of order complexes of partially ordered sets (posets). They also introduced the notion of recursive atom ordering, and they proved that a finite bounded poset is CL-shellable if and only if it admits a recursive atom ordering.
In this paper, a generalization of the notion of recursive atom ordering is introduced. A finite bounded poset is proven to admit such a generalized recursive atom ordering if and only if it admits a traditional recursive atom ordering. This is also proven equivalent to admitting a CC-shelling (a type of shelling introduced by Kozlov) with a further property called self-consistency. Thus, CL-shellability is proven equivalent to self-consistent CC-shellability. As an application, the uncrossing posets, namely the face posets for stratified spaces of planar electrical networks, are proven to be dual CL-shellable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_03949 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Generalized recursive atom ordering and equivalence to CL-shellability Hersh, Patricia Stadnyk, Grace Combinatorics 05E45, 06A07 Björner and Wachs introduced CL-shellability as a technique for studying the topological structure of order complexes of partially ordered sets (posets). They also introduced the notion of recursive atom ordering, and they proved that a finite bounded poset is CL-shellable if and only if it admits a recursive atom ordering. In this paper, a generalization of the notion of recursive atom ordering is introduced. A finite bounded poset is proven to admit such a generalized recursive atom ordering if and only if it admits a traditional recursive atom ordering. This is also proven equivalent to admitting a CC-shelling (a type of shelling introduced by Kozlov) with a further property called self-consistency. Thus, CL-shellability is proven equivalent to self-consistent CC-shellability. As an application, the uncrossing posets, namely the face posets for stratified spaces of planar electrical networks, are proven to be dual CL-shellable. |
| title | Generalized recursive atom ordering and equivalence to CL-shellability |
| topic | Combinatorics 05E45, 06A07 |
| url | https://arxiv.org/abs/2212.03949 |