Cycle Counting and Character Expectations Using Alternating Structures

[2608.27593] Cycle Counting and Character Expectations Using Alternating Structures Skip to main content Search Submit Donate Log in Mathematics > Group Theory arXiv:2608.27593 (math) [Submitted on 27 Aug 2026] Title: Cycle Counting and Character Expectations Using Alternating Structures Authors: Noam Ta Shma View PDF HTML (experimental) Abstract: Recently, two related papers [ arXiv:2412.13941 , arXiv:2409.03626 ] found a connection between two subjects: the w-cycle theorem, which is a theorem about counting appearances of cycles reading out a word w in certain graphs, and character expectations on word measures. The w-cycle theorem was proven independently by [ arXiv:1410.2540 ] using stackings and by [ arXiv:1410.2579 ] using bislim structures. In the current work, we generalize stackings and bislim structures to alternating stackings and alternating bislim structures. We show how this significantly strengthens the w-cycle theorem for words admitting such alternating structures, and as a result, also strengthens the recent results of [ arXiv:2412.13941 ] and [ arXiv:2409.03626 ]. We show that generic words admit alternating bislim structures, and therefore, the strengthened results hold for generic words. Using our new machinery, we address conjectures of Wilton, of Hanany-Puder and of Puder-Shomroni. We prove that all three conjectures hold for generic words, but we also find counterexamples for the first two. Comments: 59 pages, 22 figures Subjects: Group Theory (math.GR) ; Formal Languages and Automata Theory (cs.FL) MSC classes: 20F65, 20P05 (Primary), 60B15, 20F67(Secondary) Cite as: arXiv:2608.27593 [math.GR] (or arXiv:2608.27593v1 [math.GR] for this version) https://doi.org/10.48550/arXiv.2608.27593 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Noam Ta Shma [ view email ] [v1] Thu, 27 Aug 2026 18:26:26 UTC (230 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.GR < prev | next > new | recent | 2026-08 Change to browse by: cs cs.FL math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation 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)

Related documents

• Cycle Counting and Character Expectations Using Alternating Structures#779524
• Letter-braiding: bridging combinatorial group theory and topology#799812
• Zipf&apos;s law obtained from a fixed point in a transition in the processes of learning a language#69471
• On the individuation of words#742313
• How to measure the optimality of word or gesture order with respect to the principle of swap distance minimization#987631