User:Daablced/sandbox

The formula for the number of Young Tableaux of a given shape was originally derived from the Frobenius Determinant Formula in connection to representation theory. The formula is given by
 * $$ f(n_1,n_2,\cdots,n_m) = \frac{n! \,V_m(n_m,n_{m-1}+1,\cdots,n_1 + m - 1)}{n_m ! (n_{m-1} + 1)! \cdots (n_1 + m - 1)!}$$

where $$V_m(X_1,\cdots ,X_m)$$ is the Vandermonde determinant.

Hook lengths can also be used to give a product representation to the generating function for the number of reverse plane partitions of a given shape. If $λ$ is a partition of some integer $p$, a reverse plane partition of $n$ with shape $λ$ is obtained by filling in the boxes in the Young diagram with non-negative integers such that the entries add to $n$ and are non-decreasing along each row and down each column. The hook lengths $$ h_1,\dots,h_p $$ can be defined as with Young tableau. If $π_{n}$ denotes the number of reverse plane partitions of $n$ with shape $λ$, then the generating function can be written as
 * $$ \sum_{n=0}^\infty \pi_n x^n = \prod_{k=1}^p (1-x^{h_k})^{-1} $$