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    "title": "Pólya's cycle-index substitution theorem for weighted orbit enumeration",
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        "title": "Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Math68(1937)145–254",
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            "citation": "Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Math68(1937)145–254",
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        "description": "Finite permutation action and coefficientwise finite inventories; the series is formal and analytic convergence is not assumed. This record covers the cycle-index theorem, not every tree asymptotic or molecular example in the 110-page paper. Orbit-stabilizer/fixed-point averaging is inherited background. No exclusive priority over Redfield’s 1927 work is claimed.",
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        {
          "citation": "Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Math68(1937)145–254",
          "url": "https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/5571-11511_2006_Article_BF02546665.pdf"
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      "statement": "# Pólya's cycle-index substitution theorem for weighted orbit enumeration\n\nFor a finite permutation group acting on s positions and an inventory series f with finite nonnegative integer coefficients, the generating series of inequivalent configurations is obtained by replacing each cycle-index variable z_j with f(x^j,y^j,...). Fixed colour multiplicities are extracted coefficientwise; the proof counts cycle-constant fixed configurations and averages over the group.\n\n## Scientific scope\n\nFinite permutation action and coefficientwise finite inventories; the series is formal and analytic convergence is not assumed. This record covers the cycle-index theorem, not every tree asymptotic or molecular example in the 110-page paper. Orbit-stabilizer/fixed-point averaging is inherited background. No exclusive priority over Redfield’s 1927 work is claimed.\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\nG. Pólya\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- [Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Math68(1937)145–254](https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/5571-11511_2006_Article_BF02546665.pdf)\n",
      "summary": "For a finite permutation group acting on s positions and an inventory series f with finite nonnegative integer coefficients, the generating series of inequivalent configurations is obtained by replacing each cycle-index variable z_j with f(x^j,y^j,...). Fixed colour multiplicities are extracted coefficientwise; the proof counts cycle-constant fixed configurations and averages over the group.",
      "title": "Pólya's cycle-index substitution theorem for weighted orbit enumeration"
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    "id": "polya-finite-permutation-cycle-index-enumeration-1937",
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    "revision": 1,
    "title": "Pólya's cycle-index substitution theorem for weighted orbit enumeration",
    "context": "Enumerative combinatorics and group actions",
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    "what_changed": "For a finite permutation group acting on s positions and an inventory series f with finite nonnegative integer coefficients, the generating series of inequivalent configurations is obtained by replacing each cycle-index variable z_j with f(x^j,y^j,...). Fixed colour multiplicities are extracted coefficientwise; the proof counts cycle-constant fixed configurations and averages over the group.",
    "scope": "For a finite permutation group acting on s positions and an inventory series f with finite nonnegative integer coefficients, the generating series of inequivalent configurations is obtained by replacing each cycle-index variable z_j with f(x^j,y^j,...). Fixed colour multiplicities are extracted coefficientwise; the proof counts cycle-constant fixed configurations and averages over the group. Finite permutation action and coefficientwise finite inventories; the series is formal and analytic convergence is not assumed. This record covers the cycle-index theorem, not every tree asymptotic or molecular example in the 110-page paper. Orbit-stabilizer/fixed-point averaging is inherited background. No exclusive priority over Redfield’s 1927 work is claimed.",
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        "title": "Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Math68(1937)145–254",
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