A BIJECTION BETWEEN NONCROSSING AND NONNESTING PARTITIONS OF TYPES A, B AND C

The total number of noncrossing partitions of type Psi is equal to the nth Catalan number, 1/n+1((2n)(n)) when Psi = A(n-1), and to the corresponding binomial coefficient ((2n)(n)) when Psi = B-n or C-n. 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.