A survey of the Fine numbers

The Fine numbers and the Catalan numbers are intimately related. Two manifestations are the identity C-n = 2F(n) + Fn-1, n greater than or equal to 1, and the generating function identities F = C/(1 + zC), C = F/ (1 - zF). In this paper we collect and organize the previous literature, present many new settings, and develop the theory and generating functions as well as asymptotics. Among the topics developed are the hill-killer involution, the set of all path pairs, and some new results about noncrossing partitions. (C) 2001 Elsevier Science B.V. All rights reserved.