What is recorded here?
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.
Source-linked catalogue note. A detailed explanatory guide has not yet been prepared for this entry.
Scientific scope
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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People and systems
G. Pólya
Source-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.
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English catalogue narrative
Pólya's cycle-index substitution theorem for weighted orbit enumeration
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.
Scientific scope
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.
Reading status
This 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.
People and systems
G. Pólya
Source-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.