A326030 - OEIS

%I #4 Jul 19 2019 07:51:02

%S 2,3,6,19,132,3578,826949

%N Number of antichains of subsets of {1..n} with different edge-sums.

%C An antichain is a finite set of finite sets, none of which is a subset of any other. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

%e The a(0) = 2 through a(3) = 19 antichains:

%e   {}    {}     {}         {}

%e   {{}}  {{}}   {{}}       {{}}

%e         {{1}}  {{1}}      {{1}}

%e                {{2}}      {{2}}

%e                {{1,2}}    {{3}}

%e                {{1},{2}}  {{1,2}}

%e                           {{1,3}}

%e                           {{2,3}}

%e                           {{1},{2}}

%e                           {{1,2,3}}

%e                           {{1},{3}}

%e                           {{2},{3}}

%e                           {{1},{2,3}}

%e                           {{2},{1,3}}

%e                           {{1,2},{1,3}}

%e                           {{1,2},{2,3}}

%e                           {{1},{2},{3}}

%e                           {{1,3},{2,3}}

%e                           {{1,2},{1,3},{2,3}}

%t stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];

%t cleqset[set_]:=stableSets[Subsets[set],SubsetQ[#1,#2]||Total[#1]==Total[#2]&];

%t Table[Length[cleqset[Range[n]]],{n,0,5}]

%Y Set partitions with different block-sums are A275780.

%Y MM-numbers of multiset partitions with different part-sums are A326535.

%Y The covering case is A326572.

%Y Antichains with equal edge-sums are A326574.

%Y Cf. A000372, A003182, A006126, A321469, A326519, A326571, A326573.

%K nonn,more

%O 0,1

%A _Gus Wiseman_, Jul 18 2019