{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27593",
  "author_total": 1,
  "authors": [
    {
      "name": "Noam Ta Shma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45tpmb6gt1r9g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27593",
  "fulltext_url": null,
  "id": "45tpmb6gt1r9g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27593",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.FL"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7204855237",
  "pdf_url": "https://arxiv.org/pdf/2608.27593",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45tpmb6gt1r9g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.27593",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27593"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Cycle Counting and Character Expectations Using Alternating Structures",
  "title_text": "Cycle Counting and Character Expectations Using Alternating Structures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/45tpmb6gt1r9g",
  "venue": "arXiv",
  "versions": []
}