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A322441
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Number of pairs of set partitions of {1,...,n} where no block of one is a subset or equal to any block of the other.
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9
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OFFSET
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0,5
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COMMENTS
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For any pair (X,Y) meeting the requirement, so does the pair (Y,X) which must be distinct from (X,Y), except for X = Y = {} when n = 0. Therefore all a(n) are even for n > 0. - M. F. Hasler, Dec 30 2020
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LINKS
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EXAMPLE
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The a(4) = 6 pairs of set partitions:
{{1,2},{3,4}} and {{1,3},{2,4}},
{{1,2},{3,4}} and {{1,4},{2,3}},
{{1,3},{2,4}} and {{1,2},{3,4}},
{{1,3},{2,4}} and {{1,4},{2,3}},
{{1,4},{2,3}} and {{1,2},{3,4}},
{{1,4},{2,3}} and {{1,3},{2,4}}.
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MATHEMATICA
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sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
stabQ[u_]:=stabQ[u, SubsetQ]; stabQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Table[Length[Select[Tuples[sps[Range[n]], 2], And[UnsameQ@@Join@@#, stabQ[Join@@#]]&]], {n, 6}]
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CROSSREFS
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Cf. A000110, A000258, A001247, A008277, A059849, A060639, A181939, A318393, A321760 (unlabeled version), A322435, A322442.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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