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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 g 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 {\displaystyle f_{n}} of the number of different configurations under this re-labelling process is given by
In particular, if G is the symmetric group of order n , the functions f n {\displaystyle f_{n}} can be further combined into a single generating function:
which is exponential w.r.t. the variable z and ordinary w.r.t. the variable t.