Let n be a fixed positive integer and let An = {0, 1, ..., n) ⊆ N.
(a) How many edges are there in the Hasse diagram for the total order (An, ≤), where "≤" is the ordinary "less than or equal to" relation?
(b) In how many ways can the edges in the Hasse diagram of part (a) be partitioned so that the edges in each cell (of the partition) provide a path (of one or more edges)?
(c) In how many ways can the edges in the Hasse diagram for (A12, ≤) be partitioned so that the edges in each cell (of the partition) provide a path (of one or more edges) and one of the cells is {(3, 4), (4, 5), (5, 6), (6, 7)}?