{
  "schema_version": 1,
  "problem_number": "AMR-030-0046",
  "title": "Exact Counting and Uniform Sampling of Pressing Sequences on Paths and Cycles",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give an exact polynomial-time algorithm for counting successful pressing sequences of every labeled bicolored graph whose underlying graph has maximum degree at most two. Paths, cycles, arbitrary colorings, singular matrices, full-length sequences and disconnected unions are included. A determinant-gated last-vertex recurrence on path intervals, together with nullity bounds, yields O(n^3) arithmetic operations on O(1+n log(n+1))-bit integers. Exact uniform sampling follows from deterministic unranking in O(n^2) arithmetic operations after a uniform integer draw; fair-bit sampling has expected polynomial bit time. This answers the full-length path-counting alternative in Miklos, Dagstuhl Reports 13(3) (2023), section 4.8. General-graph hardness/approximation questions in AMR-030-0046 and rapid mixing of the earlier local-edit chain remain unresolved here. AI-assisted, self-audited preprint; no independent review, formal verification or absolute-priority certification is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "cs.DS",
    "cs.DM"
  ],
  "keywords": [
    "AMR-030-0046",
    "pressing sequences",
    "exact counting",
    "uniform sampling",
    "paths and cycles",
    "binary matrices"
  ],
  "manuscript_version_date": "2026-10-25",
  "publication_date": "2026-10-25",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-25",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-030-0046/",
  "pdf_url": "https://eulersolve.org/papers/amr-030-0046/paper.pdf?v=dc3ebc75fe08",
  "doi": "10.5281/zenodo.23642927",
  "zenodo_record_url": "https://zenodo.org/records/23642927",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Exact counting and exact uniform sampling of successful pressing sequences for every labeled bicolored graph of underlying maximum degree at most two, including singular matrices, full-length sequences, paths, cycles and disjoint unions. Preprocessing uses O(n^3) arithmetic operations on O(1+n log(n+1))-bit integers; deterministic unranking uses O(n^2) arithmetic operations. Fair-bit exact sampling has expected polynomial bit time. This answers the full-length path-counting alternative in Miklos, Dagstuhl Reports 13(3) (2023), section 4.8, but not the general-graph questions in AMR-030-0046 or rapid mixing of the previously studied Markov chain. The theorem covers labeled bicolored graphs of underlying maximum degree at most two, including singular matrices and both full-length and rank-length successful sequences. It answers the separately stated 2023 path-counting alternative. General-graph hardness/approximation and rapid mixing of the previous Markov chain are not resolved. 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": "dc3ebc75fe08f46574f8dbe63175b283c7693f4c82dd04fcd23f2235c2f641ac"
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    "source.zip": {
      "sha256": "6c59bfb270b1138302700ab2fc4fc4d7f7198e573fa376fd8fd8a18e2da136e6"
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    "verification_report.md": {
      "sha256": "dd24c3440cd1c575688675249d1388f42583a1d0b0bcd633b49be1de7ba208af"
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  },
  "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."
}