Further Combinatorial Properties of Two Fibonacci Lattices

In an earlier paper on differential posets, two lattices Fib(r) and Z(r) were defined for each positive integer r, and were shown to have some interesting combinatorial properties. In this paper the investigation of Fib(r) and Z(r) is continued. A bijection 1Jf: Fib(r)--+ Z(r) is shown to preserve many properties of the lattices, though IJf is not an isomorphism. As a consequence we give an explicit formula which generalizes the rank generating function of Fib(r) and of Z(r) . Some additional properties of Fib(r) and Z(r) are developed related to the counting of chains.