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

The total number of noncrossing partitions of type Ψ is equal to the nth Catalan number 1 n+1 ( 2n n ) when Ψ = An−1, and to the corresponding binomial coefficient ( 2n n ) when Ψ = Bn or Cn. These numbers coincide with the corresponding 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.