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    "title": "de Bruijn counts binary cycles containing every length-n word once",
    "summary": "The number of cyclic binary sequences of length 2^n containing each binary word of length n exactly once is 2^(2^(n-1)-n). The proof relates these cycles to complete directed walks and shows that doubling a two-in/two-out network with m vertices multiplies its cyclic complete-walk count by 2^(m-1).",
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    "area": "Enumerative combinatorics and directed graphs",
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    "year": 1946,
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        "title": "A combinatorial problem, Proc KNAW49(1946)758–764",
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      "scope": {
        "description": "Cyclic rotations are identified, but reversal is not identified; changing these conventions changes the count. The binary count resolves the conjecture explicitly credited to K. Posthumus. It is not a source-backed claim of first discovery of all universal cycles, nor the general-alphabet formula or full later BEST theorem.",
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          "citation": "A combinatorial problem, Proc KNAW49(1946)758–764",
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      "statement": "# de Bruijn counts binary cycles containing every length-n word once\n\nThe number of cyclic binary sequences of length 2^n containing each binary word of length n exactly once is 2^(2^(n-1)-n). The proof relates these cycles to complete directed walks and shows that doubling a two-in/two-out network with m vertices multiplies its cyclic complete-walk count by 2^(m-1).\n\n## Scientific scope\n\nCyclic rotations are identified, but reversal is not identified; changing these conventions changes the count. The binary count resolves the conjecture explicitly credited to K. Posthumus. It is not a source-backed claim of first discovery of all universal cycles, nor the general-alphabet formula or full later BEST theorem.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nN. G. de Bruijn\n\nSource-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.\n\n\n\n## Original sources\n\n- [A combinatorial problem, Proc KNAW49(1946)758–764](https://pure.tue.nl/ws/portalfiles/portal/4442708/597473.pdf)\n",
      "summary": "The number of cyclic binary sequences of length 2^n containing each binary word of length n exactly once is 2^(2^(n-1)-n). The proof relates these cycles to complete directed walks and shows that doubling a two-in/two-out network with m vertices multiplies its cyclic complete-walk count by 2^(m-1).",
      "title": "de Bruijn counts binary cycles containing every length-n word once"
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    "id": "debruijn-binary-universal-cycle-enumeration-1946",
    "entry_id": "debruijn-binary-universal-cycle-enumeration-1946",
    "revision": 1,
    "title": "de Bruijn counts binary cycles containing every length-n word once",
    "context": "Enumerative combinatorics and directed graphs",
    "track": "Exact theorem and reusable mathematical method",
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    "what_changed": "The number of cyclic binary sequences of length 2^n containing each binary word of length n exactly once is 2^(2^(n-1)-n). The proof relates these cycles to complete directed walks and shows that doubling a two-in/two-out network with m vertices multiplies its cyclic complete-walk count by 2^(m-1).",
    "scope": "The number of cyclic binary sequences of length 2^n containing each binary word of length n exactly once is 2^(2^(n-1)-n). The proof relates these cycles to complete directed walks and shows that doubling a two-in/two-out network with m vertices multiplies its cyclic complete-walk count by 2^(m-1). Cyclic rotations are identified, but reversal is not identified; changing these conventions changes the count. The binary count resolves the conjecture explicitly credited to K. Posthumus. It is not a source-backed claim of first discovery of all universal cycles, nor the general-alphabet formula or full later BEST theorem.",
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        "title": "A combinatorial problem, Proc KNAW49(1946)758–764",
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