A simple and direct derivation for the number of noncrossing partitions

Abstract

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