Bijectivity of a generalized Pak-Stanley labeling
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917362300616704 |
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| author | Bernardi, Olivier Goregaokar, Neha |
| author_facet | Bernardi, Olivier Goregaokar, Neha |
| contents | The Pak-Stanley labeling is a bijection between the regions of the $m$-Shi arrangement and the $m$-parking functions. Mazin generalized this labeling to every deformation of the braid arrangement and proved that this labeling is always surjective onto a set of directed multigraph parking functions. We provide a right inverse to the generalized Pak-Stanley labeling, and identify a class $\mathcal{C}$ of arrangements for which this labeling is bijective. The class $\mathcal{C}$ includes the multi-Shi arrangements and the multi-Catalan arrangements. We also show that the arrangements in $\mathcal{C}$ are the only transitive arrangements for which the generalized Pak-Stanley labeling is bijective. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24886 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bijectivity of a generalized Pak-Stanley labeling Bernardi, Olivier Goregaokar, Neha Combinatorics The Pak-Stanley labeling is a bijection between the regions of the $m$-Shi arrangement and the $m$-parking functions. Mazin generalized this labeling to every deformation of the braid arrangement and proved that this labeling is always surjective onto a set of directed multigraph parking functions. We provide a right inverse to the generalized Pak-Stanley labeling, and identify a class $\mathcal{C}$ of arrangements for which this labeling is bijective. The class $\mathcal{C}$ includes the multi-Shi arrangements and the multi-Catalan arrangements. We also show that the arrangements in $\mathcal{C}$ are the only transitive arrangements for which the generalized Pak-Stanley labeling is bijective. |
| title | Bijectivity of a generalized Pak-Stanley labeling |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.24886 |