Labelled enumeration theorem
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 of the number of different configurations under this re-labelling process is given by
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- enIn 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 of the number of different configurations under this re-labelling process is given by
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- enIn 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 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 (hence, |G| = n!), the functions 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.
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- Labelled enumeration theorem
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- enLabelled enumeration theorem
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- Category:Articles containing proofs
- Category:Enumerative combinatorics
- Category:Theorems in combinatorics
- Combinatorics
- Exponential generating function
- File:RelabelCycles.png
- Generating function
- Pólya enumeration theorem
- Symmetric group
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- 4qCy5
- Labelled enumeration theorem
- m.0cv8cv
- Q6466909
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- Category:Articles containing proofs
- Category:Enumerative combinatorics
- Category:Theorems in combinatorics
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