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

Abstract

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 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.

11 pages, 11 figures. Inverse map described. Minor changes to correct typos and clarify content

References in corpus (1)

• Distribution of crossings, nestings and alignments of two edges in matchings and partitions

Cited by in corpus (1)

• A bijection between noncrossing and nonnesting partitions for classical reflection groups