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Number of antichains of subsets of {1..n} with equal edge-sums.
6

%I #6 Aug 13 2019 15:44:57

%S 2,3,5,10,22,61,247,2096,81896,52260575

%N Number of antichains of subsets of {1..n} with equal 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(4) = 22 antichains:

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

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

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

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

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

%e {{1,2}} {{4}}

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

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

%e {{1,2,3}} {{1,4}}

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

%e {{2,4}}

%e {{3,4}}

%e {{1,2,3}}

%e {{1,2,4}}

%e {{1,3,4}}

%e {{2,3,4}}

%e {{1,2,3,4}}

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

%e {{4},{1,3}}

%e {{1,4},{2,3}}

%e {{2,4},{1,2,3}}

%e {{3,4},{1,2,4}}

%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 equal block-sums are A035470.

%Y Antichains with different edge-sums are A326030.

%Y MM-numbers of multiset partitions with equal part-sums are A326534.

%Y The covering case is A326566.

%Y Cf. A000372, A003182, A006126, A307249, A321455, A321717, A321718, A326518, A326565, A326572.

%K nonn,more

%O 0,1

%A _Gus Wiseman_, Jul 18 2019

%E a(9) from _Andrew Howroyd_, Aug 13 2019