The Polya Enumeration Theorem counts the number of distinct configurations of objects, such as colorings of a shape's vertices, once configurations related by a specified group of symmetries are treated as identical, by evaluating a generating-function expression called the cycle index of that symmetry group. Named for George Polya, who built it on the earlier Burnside's Lemma, it is a foundational tool of combinatorial enumeration used, for example, to count distinct necklaces, colorings of a cube's faces, or chemical isomers related by molecular symmetry.
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Partially Attested
Proof YearFirst published by Redfield in 1927 and independently rediscovered by Polya in 1937; 1927 given as first publication. Classification
Statement Form Polya Enumeration Theorem
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Source Polya enumeration theorem (Wikipedia)
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Source Polya enumeration theorem (Wikipedia)
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Source Polya enumeration theorem (Wikipedia)
Sources
1. Polya enumeration theorem (Wikipedia)
History
The theorem was first published by J. Howard Redfield in 1927.
Principal Result Of: George Pólya, Lead sentence of the article on the result
In 1937 it was independently rediscovered by George Pólya, who then greatly popularized the result by applying it to many counting problems, in particular to the enumeration of chemical compounds.
- In Branch: Combinatorics, Lead sentence
Proved By: George Pólya, Lead paragraph
The Pólya enumeration theorem, also known as the Redfield-Pólya theorem and Pólya counting, is a theorem in combinatorics that both
In Group: Enumerative, Additive and Algebraic Combinatorics, Lead section
Redfield-Pólya theorem and Pólya counting, is a theorem in combinatorics that both follows from and ultimately generalizes
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