Fibonacci numbers, ordered partitions, and transformations of a finite set

A method of representing transformations of a finite set pictorially is described. These pictures of a function are used to count certain idempotent transformations, and interesting formulae for the Fibonacci numbers are obtained. No part of any string extends beyond the region ° : : ; z ::; 1, and if two strings intersect then they join together, from the point of intersection, to meet the same terminal node. Two vines are considered equivalent if one can be continuously deformed to the other, so that at each stage of the deformation, the strings form a vine. Figure 1: A vine representing (1 f-+ 3,2 f-+ 1,3 It 3,4 It 2). To compose the vines u and v, shrink them so that u lies in the region ~ z ::; 1 and v is contained in ° : : ; z ::; ~. Identify the terminal nodes of u with the initial nodes of v. The ph string of uv is obtained by following the ph string of u until it meets a string of v and then following this string to a terminal node of v. With this composition, the equivalence classes of vines with n strings form a monoid, studied more fully in [2].