A bijection between noncrossing and nonnesting partitions of types A and B

Ricardo Mamede

The total number of noncrossing partitions of type $\Psi$ is the $n$th Catalan number $\frac{1}{n+1}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$ when $\Psi =A_{n-1}$, and the binomial $\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$ when $\Psi =B_n$, and…