A SIMPLE AND DIRECT DERIVATION FOR THE NUMBER OF NONCROSSING PARTITIONS

Kreweras considered the problem of counting noncrossing partitions of the set {1, 2, ..., n}, whose elements are arranged into a cycle in it s natural order, into p parts of given sizes n(1), n(2), ..., n(p) (bu t not specifying which part gets which size). He gave a beautiful and surprising result whose proof resorts to a recurrence relation. In thi s paper we give a direct, entirely bijective, proof starting from the same initial idea as Kreweras' proof.