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A227414
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Number of ordered n-tuples of subsets of {1,2,...,n} which satisfy the conditions in Hall's Marriage Problem.
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5
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OFFSET
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0,3
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COMMENTS
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In a group of n women and n men, each woman selects a subset of men that she would happily marry. Hall's marriage problem gives the conditions on the subsets so that every woman can become happily married.
a(n)/2^(n^2) is the probability that if the subsets are selected at random then all the women can be happy.
Equivalently, a(n) is the number of n x n {0,1} matrices such that if in any arbitrarily selected r rows we note the columns that have at least one 1 in the selected rows then the number of such columns must not be less than r.
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LINKS
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FORMULA
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EXAMPLE
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a(2) = 7 because we have:
1: ({1}, {2});
2: ({1}, {1,2});
3: ({2}, {1});
4: ({2}, {1,2});
5: ({1,2}, {1});
6: ({1,2}, {2});
7: ({1,2}, {1,2}).
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MATHEMATICA
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f[list_]:=Apply[And, Flatten[Table[Map[Length[#]>=n&, Map[Apply[Union, #]&, Subsets[list, {n}]]], {n, 1, Length[list]}]]]; Table[Total[Boole[Map[f, Tuples[Subsets[n], n]]]], {n, 1, 4}]
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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