A bijection between noncrossing and nonnesting partitions of types A and B - Mathematics > Combinatorics

Abstract: The total number of noncrossing partitions of type $\Psi$ is the $n$thCatalan number $\frac{1}{n+1}\binom{2n}{n}$ when $\Psi=A {n-1}$, and thebinomial $\binom{2n}{n}$ when $\Psi=B n$, and these numbers coincide with thecorrespondent number of nonnesting partitions.
For type A, there are severalbijective proofs of this equality, being the intuitive map that locallyconverts each crossing to a nesting one of them.
In this paper we present abijection between nonnesting and noncrossing partitions of types A and B thatgeneralizes the type A bijection that locally converts each crossing to anesting.

Author: Ricardo Mamede

Source: https://arxiv.org/