{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R199",
  "content_hash": null,
  "slug": "dclp-claim-asymptotic-expansion",
  "type": "claim",
  "title": "The probabilities tend to Euler's constant exponential",
  "summary": "Published analysis gives q_n = e^(-gamma)(1+1/n)+O(log(n)/n^2) and a full expansion.",
  "relevance": "For Eventual decrease for distinct cycle lengths in random permutations, record dclp-claim-asymptotic-expansion (“The probabilities tend to Euler's constant exponential”) records a bound, answer, status fact, or structural consequence. The record states: Published analysis gives q_n = e^(-gamma)(1+1/n)+O(log(n)/n^2) and a full expansion.",
  "relevance_source": "recorded",
  "body": "Greene and Knuth obtained\n\\[\nq_n=e^{-\\gamma}\\left(1+\\frac1n\\right)+O\\left(\\frac{\\log n}{n^2}\\right).\n\\]\nFlajolet, Fusy, Gourdon, Panario, and Pouyanne derive a full expansion with logarithmic terms and periodic contributions caused by roots of unity. In particular \\(q_n\\to e^{-\\gamma}\\). A one-step monotonicity proof needs an explicit remainder bound after differencing, since the leading predicted difference has order \\(n^{-2}\\). The sources inspected here do not supply such a bound with a threshold.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "q_n as n tends to infinity"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/math/0606370",
      "locator": "Philippe Flajolet et al., A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, Electronic Journal of Combinatorics 13 (2006), R103, Proposition 1; D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms, 2nd ed., 1982, pp. 52-54"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/math/0606370",
    "locator": "Philippe Flajolet et al., A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, Electronic Journal of Combinatorics 13 (2006), R103, Proposition 1; D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms, 2nd ed., 1982, pp. 52-54"
  },
  "models": [],
  "relations": [
    {
      "slug": "dclp-problem-eventual-strict-decrease",
      "title": "Does the distinct-cycle-length probability decrease after n=30?",
      "object_type": "problem",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "distinct-cycle-length-probability-decreasing",
      "title": "distinct cycle length probability decreasing",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}