### A bijection between noncrossing and nonnesting partitions of types A, B and C

#### 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 coefficient binomial $\binom{2n}{n}$ when $\Psi=B_n$ or $C_n$, and these numbers coincide with the

correspondent number of nonnesting partitions. For type A, there are several

bijective proofs of this equality; in particular, the intuitive map, which locally

converts each crossing to a nesting, is one of them.

In this paper we present a bijection between nonnesting and noncrossing partitions of

types $A, B$ and $C$ that generalizes the type $A$ bijection that locally converts each crossing

to a nesting.

$\frac{1}{n+1}\binom{2n}{n}$ when $\Psi=A_{n-1}$,

and the coefficient binomial $\binom{2n}{n}$ when $\Psi=B_n$ or $C_n$, and these numbers coincide with the

correspondent number of nonnesting partitions. For type A, there are several

bijective proofs of this equality; in particular, the intuitive map, which locally

converts each crossing to a nesting, is one of them.

In this paper we present a bijection between nonnesting and noncrossing partitions of

types $A, B$ and $C$ that generalizes the type $A$ bijection that locally converts each crossing

to a nesting.

#### Full Text:

PDFContributions to Discrete Mathematics. ISSN: 1715-0868