User:Bbloodaxe/sandbox

=Fuss-Catalan Explanation= The Fuss-Catalan represents the number of legal permutations or allowed ways of arranging a number of articles, that is restricted in some way. This means that they are related to the Binomial Coefficient. The key difference between Fuss-Catalan and the Binomial Coefficient is that there are no "illegal" arrangement permutations within Binomial Coefficient, but there are within Fuss-Catalan. This means that one can simply make the following equality.

All_Permutations = Legal_Permutations + Illegal_Permutations


 * $$ {mp+r \choose m} = {r \over mp+r}{mp+r \choose m} + {mp \over mp+r} {mp+r \choose m}$$


 * All_Permutations is the Binomial Coefficients
 * Legal_Permutations are the Fuss-Catalan Numbers
 * Illegal_Permutations is the remainder (a related, but different series to Fuss-Catalan)

Examples of this are Balanced brackets (see Dyck language), How many ways can 6 brackets be legally arranged? From the Binomial interpretation there are $$\tbinom 63$$ = 20 ways.

Generating Function
Let the ordinary generating function with respect to the index be defined as follows


 * $$B_{p,r}(z):=\sum_{m=0}^\infty A_m(p,r)z^m$$, then the Wojciech Mlotkowski paper (see references), shows that as
 * $$A_{m+1}(p,1)= A_{m}(p,p)$$ then it directly follows that $$B_{p,1}(z) = 1+zB_{p,p}(z)$$.

This can extended by using Lambert's equivalence $$B_{p,1}(z)^r=B_{p,r}(z)$$ to the general generating function, for all the Fuss-Catalan numbers:
 * $$B_{p,r}(z) = [1+zB_{p,r}(z)^{p/r}]^r$$.

An immediate consequence of the representation as a Gamma Function ratio is
 * $$A_m(p,p)=A_{m+1}(p,1)$$.

Then the Wojciech Mlotkowski paper (see references), shows that $$B_{p,1}(z) = 1+zB_{p,p}(z)$$. This can extended by using Lambert's equivalence $$B_{p,1}(z)^r=B_{p,r}(z)$$ to:


 * $$B_{p,r}(z) = [1+zB_{p,r}(z)^{p/r}]^r$$.

, then the Wojciech Mlotkowski paper (see references), shows that as
 * $$A_{m+1}(p,1)= A_{m}(p,p)$$ then it directly follows that $$B_{p,1}(z) = 1+zB_{p,p}(z)$$.

This can extended by using Lambert's equivalence $$B_{p,1}(z)^r=B_{p,r}(z)$$ to the general generating function, for all the Fuss-Catalan numbers: