{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R586",
  "content_hash": null,
  "slug": "pnc-attempt-prior-art-audit",
  "type": "attempt",
  "title": "The counts and recurrence are classical Hertzsprung material",
  "summary": "OEIS A002464, Riordan's 1965 paper, and Analytic Combinatorics cover the enumeration; the focused search found no explicit probability-monotonicity result.",
  "relevance": "For Monotonicity of consecutive-adjacency avoidance in random permutations, record pnc-attempt-prior-art-audit (“The counts and recurrence are classical Hertzsprung material”) documents a concrete method, search boundary, or failed route. The record states: OEIS A002464, Riordan's 1965 paper, and Analytic Combinatorics cover the enumeration; the focused search found no explicit probability-monotonicity result.",
  "relevance_source": "recorded",
  "body": "The initial counts identify OEIS A002464. Its definition matches the candidate exactly: permutations of length \\(n\\) without rising or falling successions. The entry records Hertzsprung's problem, the four-term recurrence, the inclusion-exclusion formula, the generating function, and references reaching back to the early twentieth century.\n\nJohn Riordan's 1965 paper is devoted to the recurrence. Abramson and Moser studied the broader avoidance of rising or falling \\(w\\)-sequences in 1967. Flajolet and Sedgewick derive the generating function on page 373 of `Analytic Combinatorics`. Claesson's 2022 paper places the problem in the modern theory of Hertzsprung patterns.\n\nThe ménage search led to a different family: straight ménage permutations restrict the value allowed at each position. The present condition restricts neighboring values in one-line notation. Searches using the exact initial terms, A002464, Hertzsprung, successions, normalized probabilities, and monotonicity found the classical enumeration and asymptotic ratio \\(p_n\\sim e^{-2}\\). They yielded no explicit proof or statement that \\(p_{n+1}>p_n\\) for every \\(n\\geq4\\). The monotonicity proof in this entry should therefore be treated as a new derivation with novelty unverified.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "classical and modern sources for the consecutive-value adjacency avoidance problem"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://oeis.org/A002464",
      "locator": "OEIS A002464 definition, formulas, and bibliography; searches performed 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://oeis.org/A002464",
    "locator": "OEIS A002464 definition, formulas, and bibliography; searches performed 2026-07-24"
  },
  "relations": [
    {
      "slug": "R587",
      "title": "Inclusion-exclusion gives the classical Hertzsprung numbers",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "R588",
      "title": "The avoidance probability is strictly increasing for every n at least 4",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "R1817",
      "title": "The counts and recurrence are classical Hertzsprung material",
      "object_type": "attempt",
      "relation": "supersedes",
      "direction": "incoming"
    },
    {
      "slug": "permutation-no-consecutive-adjacency-monotone",
      "title": "permutation no consecutive adjacency monotone",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}