Parts (a) and (b) of Fig. 10.27 provide the Hasse diagrams for two partial orders referred to

Question:

Parts (a) and (b) of Fig. 10.27 provide the Hasse diagrams for two partial orders referred to as thq fences F5, F6 [on 5, 6 (distinct) elements, respectively]. If, for instance, R denotes the partial order for the fence F5, then a1 R a2, a3 R a2, a3 R a4, and a5 R a4. For each such fence Fn, n ‰¥ 1, we follow the convention that an element with an odd subscript is minimal and one with an even subscript is maximal. Let ({1, 2}, ‰¤ ) denote the partial order where ‰¤ denotes the usual "less than or equal to" relation. As in Exercise 26 of Section 7.3, a function f: fn †’ {1, 2} is called order-preserving when for all x, y ˆˆ, x R y †’ f(x) ‰¤ f(y). Let cn count the number of such order-preserving functions. Find and solve a recurrence relation for cn.