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

The total number of noncrossing partitions of type Ψ is the nth Catalan number 1n+1(2nn) when Ψ=An−1, and the coefficient binomial (2nn) when Ψ=Bn or Cn, 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.