Reliability and efficiency are important criteria in the design of interconnection networks. Recently, the w-wide diameter d w (G), the (w − 1)-fault diameter D w (G), and the w-Rabin number r w (G) have been used to measure network reliability and efficiency. In this paper, we study wide diameters for an important class of parallel networksbutterfly networks. The main result of this paper is to determine their wide diameters.

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 , · · · , np (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.

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