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.