A Bijection between Maximal Chains in Fibonacci Posets

In a 1988 paper [9], Stanley introduced a class of partially ordered sets, called differential posets, defined independently by S. Fomin [1] who called them Y-graphs. The prototypical example of a differential poset is Young's lattice, Y, the lattice of integer partitions ordered by inclusion of Ferrers diagrams. Another important example is given by the Fibonacci r-differential poset, Z(r), defined for each positive integer r. In [7, 9] Stanley introduced the partially ordered set Fib(r) which has the same elements as Z(r), but different covering relations. He showed that: (a) for both posets, the number of pairs of saturated chains from the smallest element, 0 , to elements, w, of height n is rn !; (b) for every element w, Fib(r) and Z(r) afford the same number of maximal chains in the interval [0 , w]; and in [10] that (c) they have the same number of elements in each such interval. Stanley's proofs of these results are algebraic. Statement (b) was also shown by Fomin [1]. In [2, 4], Fomin gave a constructive proof of (a). In this paper, we will give a bijective proof of (b). A constructive proof of (c) can be found in [6]. This bijection relies on making explicit the significance of individual columns of certain tableaux defined by Stanley, Fomin, and Kemp [7, 2, 5] with respect to the corresponding chains. A simplification of these tableaux article no. TA972764