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A322441
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.
9
1, 0, 0, 0, 6, 60, 630, 9660, 192906
OFFSET
0,5
COMMENTS
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
EXAMPLE
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}}.
MATHEMATICA
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}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 08 2018
STATUS
approved