Answer Happy

For a set A, let PA be the set of partitions of A. A partition R = {R₁, R₂, ..., Rn} is a refinement of a partition P = {P₁, P2,..., Pm} if for every R, ER, there exists P; EP such that R₁ C P₁. For example, for the set A = {1, 2, 3, 4, 5), we consider the three partitions W = {{1,2}, {4}, {3,5}}, X = {{1,2}, {3, 4, 5}}, and Y = {{1, 2, 3}, {4,5}}. Now W is a refinement of X, but W is not a refinement of Y. Let R be a relation on PA that satisfies RRP if and only if R is a refinement of P. (a) Prove that for any set A, R is a partial order on PA. (b) Give the values of the Stirling numbers S(4, 1), S(4,2), S(4, 3) and S(4, 4). How many distinct partitions are there of a set with 4 elements? (c) If A = {1, 2, 3, 4), draw the Hasse diagram for the partial order R on PA. For part (c), you may use a condensed notation to avoid a lot of set brackets by listing element with a dash to indicate a new part. For the examples above, we would write W as 12-4-35, X as 12-345 and Y as 123-45.