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COMMENTS
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There is the following general formula: The number T(n,k,r) of n-tuples where each entry is chosen from the subsets of {1,2,..,k} such that the intersection of all n entries contains exactly r elements is: T(n,k,r) = C(k,r) * (2^n - 1)^(k-r). This may be shown by exhibiting a bijection to a set whose cardinality is obviously C(k,r) * (2^n - 1)^(k-r), namely the set of all k-tuples where each entry is chosen from subsets of {1,..,n} in the following way: Exactly r entries must be {1,..,n} itself (there are C(k,r) ways to choose them) and the remaining (k-r) entries must be chosen from the 2^n-1 proper subsets of {1,..,n}, i.e. for each of the (k-r) entries, {1,..,n} is forbidden (there are, independent of the choice of the full entries, (2^n - 1)^(k-r) possibilities to do that, hence the formula). The bijection into this set is given by (X_1,..,X_n) |-> (Y_1,..,Y_k) where for each j in {1,..,k} and each i in {1,..,n}, i is in Y_j if and only if j is in X_i.
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EXAMPLE
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a(1)=3 because the three sequences of length one are: ({1}), ({2}), ({3}).
a(2)=27 because the twenty-seven sequences of length two are:
({1},{1}), ({2},{2}), ({3},{3}), ({1},{1,2}),
({1},{1,3}), ({2},{1,2}), ({2},{2,3}), ({3},{1,3}),
({3},{2,3}), ({1,2},{1}), ({1,3},{1}), ({1,2},{2}),
({2,3},{2}), ({1,3},{3}), ({2,3},{3}), ({1},{1,2,3}),
({2},{1,2,3}), ({3},{1,2,3}), ({1,2,3},{1}), ({1,2,3},{2}),
({1,2,3},{3}), ({1,2},{1,3}), ({1,3},{1,2}), ({1,2},{2,3}),
({2,3},{1,2}), ({1,3},{2,3}), ({2,3},{1,3}).
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