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Fix positive integers n and k. Find the number of k-tuples (S1, S2, . . ....

Fix positive integers n and k. Find the number of k-tuples (S1, S2, . . . , Sk) of subsets Si of [n] = {1, 2, . . . , n} subject to each of the following conditions separately, that is, the three parts are independent problems.

(a) S1 ⊆ S2 ⊆ · · · ⊆ Sk.

(b) The Si are pairwise disjoint (i.e. Si ∩ Sj = ∅ for i 6= j).

(c) S1 ∩ S2 ∩ · · · ∩ Sk = ∅.

Math   Advanced-Math   
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