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

TL;DR

This paper introduces a bijection between noncrossing and nonnesting partitions of types A and B, generalizing existing type A bijections and revealing their combinatorial equivalence.

Contribution

It presents a new bijective proof connecting noncrossing and nonnesting partitions for types A and B, extending previous type A results.

Findings

Establishes a bijection between noncrossing and nonnesting partitions of types A and B.

Generalizes the type A bijection to include type B partitions.

Shows the equivalence of counts for noncrossing and nonnesting partitions in these types.

Abstract

The total number of noncrossing partitions of type $\Psi$ is the $n$th Catalan number $\frac{1}{n+1}\binom{2n}{n}$ when $\Psi=A_{n-1}$, and the binomial $\binom{2n}{n}$ when $\Psi=B_n$, and these numbers coincide with the correspondent number of nonnesting partitions. For type A, there are several bijective proofs of this equality, being the intuitive map that locally converts each crossing to a nesting one of them. In this paper we present a bijection between nonnesting and noncrossing partitions of types A and B that generalizes the type A bijection that locally converts each crossing to a nesting.