{
  "schema_version": 1,
  "problem_number": "OWR-14299904-011",
  "title": "Exact Enumeration of Word Trees with a Singleton Letter",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give exact counts of word trees whenever the defining word contains a letter exactly once. A word tree is an assignment of independent uniform transition maps whose composite has exactly one cyclic point. Conditioning on the singleton-letter map reduces the count to a two-point collision probability for the remaining word. A finite recursion on equality patterns of two trajectories computes this probability exactly, with a state space depending on the word rather than n. For two maps, T_AB(n)=(2n-1)n^(2n-2) and T_AAB(n)=(3n^3-3n^2-n+2)n^(2n-4), for n>=2; all A^rB words, rotations and positive powers are covered. The five-page proof credits the classical weighted Cayley/Pitman mapping identity and Martinsson's conditioning construction. These formulas answer Chapuy's explicitly proposed AB and AAB cases. The general enumeration, total-variation questions and neighboring permutation questions bundled into UnsolvedMath OWR-14299904-011 are not solved; the whole dataset record remains partially solved. Source code and exact checks accompany the paper. This is an unrefereed preprint developed with AI assistance, without formal proof-assistant verification or an absolute-priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "keywords": [
    "OWR-14299904-011",
    "word trees",
    "random automata",
    "exact enumeration",
    "collision probability",
    "weighted Cayley formula"
  ],
  "manuscript_version_date": "2026-10-27",
  "publication_date": "2026-10-27",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-27",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14299904-011/",
  "pdf_url": "https://eulersolve.org/papers/owr-14299904-011/paper.pdf?v=2ce6f233636f",
  "doi": "10.5281/zenodo.23678591",
  "zenodo_record_url": "https://zenodo.org/records/23678591",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Exact enumeration of all word trees whose word has a singleton letter, with a finite two-trajectory recursion independent of n; closed formulas T_AB=(2n-1)n^(2n-2) and T_AAB=(3n^3-3n^2-n+2)n^(2n-4). Includes every A^rB and cyclic rotations and powers. Complete exact counts for the explicitly proposed AB and AAB cases and all singleton-letter words. The general word-tree enumeration, total-variation questions and neighboring permutation conjectures remain unresolved here. Classical weighted-mapping and conditioning results are credited. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "2ce6f233636f604a41ef9b3339f9fe30721ccec1ee2f1551331c76f7e425e7b1"
    },
    "source.zip": {
      "sha256": "2324a86caa46907691e63eb42a78e1fd90571f1745215e48e27cb5928696b312"
    },
    "verification_report.md": {
      "sha256": "d092dd9660adde8f64827e235cd687e96bdc99d8acdb613f8d456f795bdc72cc"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}