---
tag: XNX30CK7S5
kind: statement
role: proposition
name: Labelled enumeration theorem
sources:
  - wikipedia: en:34871920@r1195473520
    locator: Labelled enumeration theorem
    note: adopted
    mention: wikipedia://en/34871920
    sections: [Statement]
same_as:
  wikidata: Q6466909
  wikipedia: Labelled enumeration theorem
inventory: [YAGE2H0J3X]
---

## Statement

In [[combinatorial|0JKW7W7XFR]] 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 $f_n(z)$ of the number of different configurations under this re-labelling process is given by