Labeling regions in deformations of graphical arrangements

[2312.06513] Labeling regions in deformations of graphical arrangements Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2312.06513 (math) [Submitted on 11 Dec 2023 ( v1 ), last revised 11 Apr 2026 (this version, v5)] Title: Labeling regions in deformations of graphical arrangements Authors: Gábor Hetyei View a PDF of the paper titled Labeling regions in deformations of graphical arrangements, by G\'abor Hetyei View PDF HTML (experimental) Abstract: Combining Carver's variant of the Farkas' lemma with the Flow Decomposition Theorem we show that the regions of any deformation of a graphical arrangement may be bijectively labeled with a set of weighted digraphs containing directed cycles of negative weight only. Bounded regions correspond to strongly connected digraphs. The study of the resulting labelings allows us to add the omitted details in Stanley's proof on the injectivity of the Pak-Stanley labeling of the regions of the extended Shi arrangement, to generalize the ceiling diagrams in the deleted Shi and Ish arrangements studied by Armstrong and Rhoades and to introduce a new labeling of the regions in the Fuss-Catalan arrangement. We also point out that Athanasiadis-Linusson labelings may be used to directly count regions in a class of arrangements properly containing the extended Shi arrangement and the Fuss-Catalan arrangement. Comments: Substantial corrections in Section 3, added a new section on ceiling diagrams Subjects: Combinatorics (math.CO) MSC classes: Primary 52C35, Secondary 05A10, 05A15, 11B68, 11B83 Cite as: arXiv:2312.06513 [math.CO] (or arXiv:2312.06513v5 [math.CO] for this version) https://doi.org/10.48550/arXiv.2312.06513 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Combinatorial Theory 5 (4) (2025) # 15 Related DOI : https://doi.org/10.5070/C65465681 Focus to learn more DOI(s) linking to related resources Submission history From: Gábor Hetyei [ view email ] [v1] Mon, 11 Dec 2023 16:39:19 UTC (43 KB) [v2] Wed, 20 Dec 2023 04:58:20 UTC (45 KB) [v3] Thu, 2 May 2024 09:59:19 UTC (45 KB) [v4] Mon, 26 May 2025 13:45:21 UTC (56 KB) [v5] Sat, 11 Apr 2026 04:25:14 UTC (60 KB) Full-text links: Access Paper: View a PDF of the paper titled Labeling regions in deformations of graphical arrangements, by G\'abor Hetyei View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2023-12 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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