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A318403
Number of strict connected antichains of sets whose multiset union is an integer partition of n.
2
1, 1, 1, 2, 2, 3, 4, 6, 8, 12, 13, 22, 31
OFFSET
0,4
EXAMPLE
The a(1) = 1 through a(10) = 13 clutters:
{{1}} {{2}} {{3}} {{4}} {{5}} {{6}} {{7}}
{{1,2}} {{1,3}} {{1,4}} {{1,5}} {{1,6}}
{{2,3}} {{2,4}} {{2,5}}
{{1,2,3}} {{3,4}}
{{1,2,4}}
{{1,2},{1,3}}
.
{{8}} {{9}} {{10}}
{{1,7}} {{1,8}} {{1,9}}
{{2,6}} {{2,7}} {{2,8}}
{{3,5}} {{3,6}} {{3,7}}
{{1,2,5}} {{4,5}} {{4,6}}
{{1,3,4}} {{1,2,6}} {{1,2,7}}
{{1,2},{1,4}} {{1,3,5}} {{1,3,6}}
{{1,2},{2,3}} {{2,3,4}} {{1,4,5}}
{{1,2},{1,5}} {{2,3,5}}
{{1,2},{2,4}} {{1,2,3,4}}
{{1,3},{1,4}} {{1,2},{1,6}}
{{1,3},{2,3}} {{1,2},{2,5}}
{{1,3},{1,5}}
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
submultisetQ[M_, N_]:=Or[Length[M]==0, MatchQ[{Sort[List@@M], Sort[List@@N]}, {{x_, Z___}, {___, x_, W___}}/; submultisetQ[{Z}, {W}]]];
antiQ[s_]:=Select[Tuples[s, 2], And[UnsameQ@@#, submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n], And[UnsameQ@@#, And@@UnsameQ@@@#, Length[csm[#]]==1, antiQ[#]]&]], {n, 8}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Oct 12 2018
STATUS
approved