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User:The tree stump/Fuss-Catalan number

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In combinatorial mathematics, the Fuss-Catalan numbers are a generalization of the Catalan numbers. For any non-negative integer and any well-generated complex reflection group, they form a sequence of natural numbers. Those occur - as the Catalan numbers - in the context of various counting problems.

In full generality, the Fuss-Catalan numbers are defined for an integer and a well-generated complex reflection group by

where denotes the rank of , where denote its degrees, and where denotes its Coxeter number.

The Fuss-Catalan numbers are named after the Belgian mathematician Eugène Charles Catalan (1814–1894) and after the Swiss mathematician Nicolas Fuss (1755–1826).

Fuss-Catalan numbers for the classical groups

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The symmetric group (group of permutations)

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For the symmetric group , which is the reflection group ,

The hyperoctahedral group (group of signed permutations)

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For the hyperoctahedral group, which is the reflection group ,

Group of even-signed permutations

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For the group of even-signed permutations, which is the reflection group ,

History

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This expression which moreover reduces to the classical Catalan numbers for . Therefore, is often called classical Fuss-Catalan numbers or generalized Catalan numbers.

Applications in Combinatorics

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Fuss-Narayana numbers

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References

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Category:Integer sequences Category:Factorial and binomial topics Category:Enumerative combinatorics