Theorem

Polya Enumeration Theorem

Combinatorics and Graph Theory

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.

Facts

Partially Attested

Proof Year

1927 1

First published by Redfield in 1927 and independently rediscovered by Polya in 1937; 1927 given as first publication.

Classification

Statement Form

Identity or Equation 1

Connections

Has Statement Form

Equation

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

Identity

In Branch

Combinatorics

Source Polya enumeration theorem (Wikipedia)

Proved By

George Polya

Source Polya enumeration theorem (Wikipedia)

Sources

1. Polya enumeration theorem (Wikipedia)

History
The theorem was first published by J. Howard Redfield in 1927.
In Branch: Combinatorics, Lead sentence
Proved By: George Polya, Lead paragraph
The Pólya enumeration theorem, also known as the Redfield-Pólya theorem and Pólya counting, is a theorem in combinatorics that both

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Polya Enumeration Theorem

Has Statement Form

In Branch

Proved By