| In a 1988 paper R. Stanley introduced a class of partially ordered sets, called differential posets, defined independently by S. Fomin who called them Y-graphs. The prototypical example of a differential poset is Young's lattice. Another important example is given by the Fibonacci r-differential poset, Z(r). Many of the enumerative results related to these posets were originally consequences of representation theory and the theory of symmetric functions, but now can be shown to depend only on simple structural properties of the poset.;The partially ordered set Fib(r), also introduced by Stanley, has the same elements as Z(r) but different covering relations and does not belong to the class of differential posets. He showed that: (a) for both posets, the number of pairs of saturated chains from the minimal element, 0 to all elements of height n is ;Relying on the encoding of chains as certain tableaux developed by Fomin, Roby and Kemp, we were able to construct a bijection between maximal chains in the intermakes explicit the significance of individual columns of the tableaux with respect to the corresponding chains. As a result of this bijection, some of the algebraic tools developed by Stanley for enumerating chains in Z(r) can be explained in terms of the constructive methods applied to Fib(r). The proof that these intervals have the same number of elements uses R-labellings of Fib(r) and Z(r) to first construct a bijection between the maximal chains in (0, w) of Fib(r) whose labels satisfy the property of having exactly one descent and maximal chains in the interval (0,w) of Z(r) whose labels have exactly one descent. |
