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).

The symmetric group (group of permutations)
For the symmetric group $$\mathfrak{S}_n$$, which is the reflection group $$A_{n-1} = G(1,1,n)$$,
 * $$C^{(m)}_n := {\rm Cat}^{(m)}(A_{n-1}) = \frac{1}{mn+1}{(m+1)n\choose n}.$$

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, $$C^{(m)}_n$$ is often called classical Fuss-Catalan numbers or generalized Catalan numbers.