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%I #13 Aug 14 2019 01:48:47
%S 1,0,1,1,13,121,2566,121199,13254529
%N Number of covering antichains of subsets of {1..n} with no singletons and different edge-sizes.
%C An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sizes are the numbers of vertices in each edge, so for example the edge sizes of {{1,3},{2,5},{3,4,5}} are {2,2,3}.
%F a(n) = A326570(n) - n*a(n-1) for n > 0. - _Andrew Howroyd_, Aug 13 2019
%e The a(2) = 1 through a(4) = 13 antichains:
%e {{1,2}} {{1,2,3}} {{1,2,3,4}}
%e {{1,2},{1,3,4}}
%e {{1,2},{2,3,4}}
%e {{1,3},{1,2,4}}
%e {{1,3},{2,3,4}}
%e {{1,4},{1,2,3}}
%e {{1,4},{2,3,4}}
%e {{2,3},{1,2,4}}
%e {{2,3},{1,3,4}}
%e {{2,4},{1,2,3}}
%e {{2,4},{1,3,4}}
%e {{3,4},{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 cleq[n_]:=Select[stableSets[Subsets[Range[n],{2,n}],SubsetQ[#1,#2]||Length[#1]==Length[#2]&],Union@@#==Range[n]&];
%t Table[Length[cleq[n]],{n,0,6}]
%Y Antichain covers are A006126.
%Y Set partitions with different block sizes are A007837.
%Y The case with singletons is A326570.
%Y Cf. A000372, A003182, A306021, A307249, A326565, A326571, A326572, A326573.
%K nonn,more
%O 0,5
%A _Gus Wiseman_, Jul 18 2019
%E a(8) from _Andrew Howroyd_, Aug 13 2019
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