User:The tree stump/Fuss-Catalan number

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 $m\geq 0$ and a well-generated complex reflection group $W$ by

${\rm {Cat}}^{(m)}(W)=\prod _{i=1}^{\ell }{\frac {d_{i}+mh}{d_{i}}},$

where $\ell$ denotes the rank of $W$, where $d_{1}\leq \ldots \leq d_{\ell }$ denote its degrees, and where $h:=d_{\ell }$ 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

The symmetric group (group of permutations)

For the symmetric group ${\mathfrak {S}}_{n}$, which is the reflection group $A_{n-1}=G(1,1,n)$,

$C_{n}^{(m)}:={\rm {Cat}}^{(m)}(A_{n-1})={\frac {1}{mn+1}}{(m+1)n \choose n}.$

The hyperoctahedral group (group of signed permutations)

For the hyperoctahedral group, which is the reflection group $B_{n}=G(2,1,n)$,

${\rm {Cat}}^{(m)}(B_{n})={(m+1)n \choose n}.$

Group of even-signed permutations

For the group of even-signed permutations, which is the reflection group $D_{n}=G(2,2,n)$,

${\rm {Cat}}^{(m)}(D_{n})={(m+1)n \choose n}-{(m+1)(n-1) \choose n-1}.$

History

This expression which moreover reduces to the classical Catalan numbers $C_{n}$ for $m=1$. Therefore, $C_{n}^{(m)}$ is often called classical Fuss-Catalan numbers or generalized Catalan numbers.

External links

• example.com

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