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%I #13 Jan 02 2021 10:35:19
%S 1,0,0,0,6,60,630,9660,192906
%N 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.
%C 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
%e The a(4) = 6 pairs of set partitions:
%e {{1,2},{3,4}} and {{1,3},{2,4}},
%e {{1,2},{3,4}} and {{1,4},{2,3}},
%e {{1,3},{2,4}} and {{1,2},{3,4}},
%e {{1,3},{2,4}} and {{1,4},{2,3}},
%e {{1,4},{2,3}} and {{1,2},{3,4}},
%e {{1,4},{2,3}} and {{1,3},{2,4}}.
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t stabQ[u_]:=stabQ[u,SubsetQ];stabQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t Table[Length[Select[Tuples[sps[Range[n]],2],And[UnsameQ@@Join@@#,stabQ[Join@@#]]&]],{n,6}]
%Y Cf. A000110, A000258, A001247, A008277, A059849, A060639, A181939, A318393, A321760 (unlabeled version), A322435, A322442.
%K nonn,more
%O 0,5
%A _Gus Wiseman_, Dec 08 2018
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