Math 2.0

Proposition

Labelled enumeration theorem

Statement

In combinatorial mathematics, the labelled enumeration theorem is the counterpart of the Pólya enumeration theorem for the labelled case, where we have a set of labelled objects given by an exponential generating function (EGF) g(z) which are being distributed into n slots and a permutation group G which permutes the slots, thus creating equivalence classes of configurations. There is a special re-labelling operation that re-labels the objects in the slots, assigning labels from 1 to k, where k is the total number of nodes, i.e. the sum of the number of nodes of the individual objects. The EGF fn(z)f_n(z) of the number of different configurations under this re-labelling process is given by

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first. The path takes the proof Proof of Labelled enumeration theorem.

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Proof of Labelled enumeration theorem, proof, no evidence levelProof of Labelled enu…proofLabelled enumeration theorem, statement, no evidence levelLabelled enumeration …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Proof of Labelled enumeration theorem proof
  2. Labelled enumeration theorem statement

Further reading

  1. Wikipedia, page 34871920, revision 1195473520, Labelled enumeration theorem. adopted
About this page
TagXNX30CK7S5
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Kindstatement, proposition
WritingW0 stub
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Changed4 November 2026

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Proofs