Pólya enumeration theorem
E586574
The Pólya enumeration theorem is a fundamental result in combinatorics that counts distinct configurations of objects under group actions by using cycle index polynomials and generating functions.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Pólya enumeration theorem canonical | 2 |
| Pólya’s counting theory | 1 |
| necklace (combinatorics) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6327842 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Pólya enumeration theorem Context triple: [enumerative combinatorics, usesConcept, Pólya enumeration theorem]
-
A.
The Twelvefold Way
The Twelvefold Way is a framework in combinatorics that systematically classifies twelve fundamental ways of counting functions between finite sets under various labeling and structural constraints.
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B.
enumerative combinatorics
Enumerative combinatorics is a branch of mathematics focused on counting and characterizing discrete structures, often using generating functions, bijections, and algebraic techniques.
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C.
Foundations of Combinatorial Theory
Foundations of Combinatorial Theory is a seminal mathematical work by Gian-Carlo Rota that helped establish modern combinatorics as a rigorous and unified field of study.
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D.
Symanzik polynomials
Symanzik polynomials are graph-based polynomials that arise in the parametric representation of Feynman integrals in quantum field theory, encoding the topology and kinematic dependence of Feynman diagrams.
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E.
Sylvester’s theorem on partitions
Sylvester’s theorem on partitions is a result in number theory that provides a systematic way to count integer partitions subject to certain congruence or restriction conditions, forming part of the foundational work in partition theory.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Pólya enumeration theorem Target entity description: The Pólya enumeration theorem is a fundamental result in combinatorics that counts distinct configurations of objects under group actions by using cycle index polynomials and generating functions.
-
A.
The Twelvefold Way
The Twelvefold Way is a framework in combinatorics that systematically classifies twelve fundamental ways of counting functions between finite sets under various labeling and structural constraints.
-
B.
enumerative combinatorics
Enumerative combinatorics is a branch of mathematics focused on counting and characterizing discrete structures, often using generating functions, bijections, and algebraic techniques.
-
C.
Foundations of Combinatorial Theory
Foundations of Combinatorial Theory is a seminal mathematical work by Gian-Carlo Rota that helped establish modern combinatorics as a rigorous and unified field of study.
-
D.
Symanzik polynomials
Symanzik polynomials are graph-based polynomials that arise in the parametric representation of Feynman integrals in quantum field theory, encoding the topology and kinematic dependence of Feynman diagrams.
-
E.
Sylvester’s theorem on partitions
Sylvester’s theorem on partitions is a result in number theory that provides a systematic way to count integer partitions subject to certain congruence or restriction conditions, forming part of the foundational work in partition theory.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
combinatorial theorem
ⓘ
enumeration theorem ⓘ |
| alsoKnownAs | Pólya counting theorem NERFINISHED ⓘ |
| appliedIn |
chemical graph theory
ⓘ
coding theory ⓘ combinatorial species ⓘ counting chemical isomers ⓘ counting colorings of graphs ⓘ counting colorings of necklaces ⓘ counting colorings of polyhedra ⓘ design of experiments ⓘ |
| appliesTo |
configurations up to symmetry
ⓘ
finite group ⓘ set of colorings ⓘ |
| assumes |
finite number of colors
ⓘ
finite permutation group ⓘ |
| concerns |
equivalence classes of colorings
ⓘ
symmetry groups of combinatorial objects ⓘ |
| defines | cycle index of a permutation group ⓘ |
| expresses | count of colorings as evaluation of cycle index polynomial ⓘ |
| field |
combinatorics
ⓘ
enumerative combinatorics ⓘ |
| generalizes | Burnside's lemma NERFINISHED ⓘ |
| hasFormulation |
cycle index series formulation
ⓘ
weight inventory formulation ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| implies | number of orbits equals average of fixed colorings ⓘ |
| influenced |
modern enumerative combinatorics
ⓘ
theory of combinatorial species ⓘ |
| namedAfter | George Pólya NERFINISHED ⓘ |
| purpose | count distinct configurations under group actions ⓘ |
| relatedTo |
Redfield–Pólya theorem
NERFINISHED
ⓘ
cycle index of the symmetric group ⓘ necklace counting problem ⓘ orbit-counting theorem NERFINISHED ⓘ |
| relates |
cycle structure of permutations
ⓘ
group action on a set ⓘ number of inequivalent colorings ⓘ orbits of a group action ⓘ |
| typicalExample |
counting colorings of the faces of a cube
ⓘ
counting colorings of vertices of a regular polygon ⓘ |
| usesConcept |
Burnside's lemma
NERFINISHED
ⓘ
cycle index polynomial ⓘ generating function ⓘ group action ⓘ |
| usesOperation |
exponential generating function
ⓘ
substitution into cycle index polynomial ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Pólya enumeration theorem Description of subject: The Pólya enumeration theorem is a fundamental result in combinatorics that counts distinct configurations of objects under group actions by using cycle index polynomials and generating functions.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Pólya’s counting theory
this entity surface form:
necklace (combinatorics)