Shellability of the quotient order on lattice path matroids
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912319437537280 |
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| author | Benedetti, Carolina Dochtermann, Anton Knauer, Kolja Li, Yupeng |
| author_facet | Benedetti, Carolina Dochtermann, Anton Knauer, Kolja Li, Yupeng |
| contents | The concept of a matroid quotient has connections to fundamental questions in the geometry of flag varieties. In previous work, Benedetti and Knauer characterized quotients in the class of lattice path matroids (LPMs) in terms of a simple combinatorial condition. As a consequence, they showed that the quotient order on LPMs yields a graded poset whose rank polynomial relates to a refinement of the Catalan numbers. In this work we show that this poset admits an EL-labeling, implying that the order complex is shellable and hence enjoys several combinatorial and topological properties. We use this to establish bounds on the Möbius function of the poset, interpreting falling chains in the EL-labeling in terms of properties of underlying permutations. Furthermore, we show that this EL-labeling is in fact a Whitney labeling, in the sense of the recent notion introduced by González D'León and Hallam. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_07306 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shellability of the quotient order on lattice path matroids Benedetti, Carolina Dochtermann, Anton Knauer, Kolja Li, Yupeng Combinatorics 05B35, 52B22, 05A15, 06A07 The concept of a matroid quotient has connections to fundamental questions in the geometry of flag varieties. In previous work, Benedetti and Knauer characterized quotients in the class of lattice path matroids (LPMs) in terms of a simple combinatorial condition. As a consequence, they showed that the quotient order on LPMs yields a graded poset whose rank polynomial relates to a refinement of the Catalan numbers. In this work we show that this poset admits an EL-labeling, implying that the order complex is shellable and hence enjoys several combinatorial and topological properties. We use this to establish bounds on the Möbius function of the poset, interpreting falling chains in the EL-labeling in terms of properties of underlying permutations. Furthermore, we show that this EL-labeling is in fact a Whitney labeling, in the sense of the recent notion introduced by González D'León and Hallam. |
| title | Shellability of the quotient order on lattice path matroids |
| topic | Combinatorics 05B35, 52B22, 05A15, 06A07 |
| url | https://arxiv.org/abs/2504.07306 |