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  "slug": "cayley-weighted-fixed-points",
  "title": "Weighted fixed points of Cayley permutations",
  "shortTitle": "When position meets value",
  "url": "https://evidencepress.org/releases/cayley-weighted-fixed-points/",
  "oneLine": "A short symmetry argument proves a weighted fixed-point identity for every Cayley-word length, linking it to classical rank counts.",
  "abstract": "Unrefereed written-proof candidate resolving Cerbai, Claesson, Ernst and Golab Conjecture 7.4 for every n>=1: half the primitive Cayley-word count at n+1 equals the total letter sum at n divided by n and the weighted fixed-point sum. Classical scalar enumeration and barred arrangements are credited; the contribution is the explicit fixed-point interpretation. Includes explanatory fixed-alphabet arbitrary-weight refinement and elementary primitive recurrence. Exact internal enumeration through length eight and count checks through n=30 do not substitute for the universal proof, external peer review or formal verification.",
  "datePublished": "2026-10-12",
  "dateModified": "2026-10-12",
  "version": "1.1.0-candidate",
  "doi": "10.5281/zenodo.23314214",
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  "repoUrl": "https://github.com/ipitchford/cayley-weighted-fixed-points",
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  "authors": [
    "Anonymous"
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      "note": "Fresh-extraction exact checks enumerate Cayley words through length eight, including fixed-alphabet strata and signed weights; count formulas agree through n=30. Four false expectations are rejected under normal and optimized Python. Finite tests do not prove the all-length written theorem.",
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    "name": "CCEG Conjecture 7.4",
    "url": "https://arxiv.org/abs/2407.19583"
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  "keywords": [
    "Cayley permutations",
    "fixed points",
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    "combinatorics",
    "exchangeability"
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    "The weighted fixed-point identity holds for every n>=1.",
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    {
      "citation": "Cerbai, Claesson, Ernst and Golab (2026), Pattern-avoiding Cayley permutations via combinatorial species: Theorem 7.2 and Conjecture 7.4.",
      "url": "https://arxiv.org/abs/2407.19583",
      "doi": "10.48550/arXiv.2407.19583"
    },
    {
      "citation": "Sloane and Wieder (2004), The number of hierarchical orderings, Section 4: classical rank marginal and scalar enumerator.",
      "url": "https://arxiv.org/abs/math/0307064",
      "doi": null
    },
    {
      "citation": "Ahlbach, Usatine and Pippenger (2013), Barred preferential arrangements, Section 3: standard marked cut.",
      "url": "https://doi.org/10.37236/2482",
      "doi": "10.37236/2482"
    },
    {
      "citation": "Cerbai and Claesson (2026), Counting fixed-point-free Cayley permutations: a distinct fixed-point question.",
      "url": "https://arxiv.org/abs/2507.09304v4",
      "doi": "10.48550/arXiv.2507.09304"
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