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A327350 Triangle read by rows where T(n,k) is the number of antichains of nonempty sets covering n vertices with vertex-connectivity >= k.

%I A327350 #14 May 25 2021 01:40:58
%S A327350 1,1,0,2,1,0,9,5,2,0,114,84,44,17,0,6894,6348,4983,3141,1451,0,7785062
%N A327350 Triangle read by rows where T(n,k) is the number of antichains of nonempty sets covering n vertices with vertex-connectivity >= k.
%C A327350 An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.
%C A327350 The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.
%C A327350 If empty edges are allowed, we have T(0,0) = 2.
%e A327350 Triangle begins:
%e A327350      1
%e A327350      1    0
%e A327350      2    1    0
%e A327350      9    5    2    0
%e A327350    114   84   44   17    0
%e A327350   6894 6348 4983 3141 1451    0
%e A327350 The antichains counted in row n = 3:
%e A327350   {123}         {123}         {123}
%e A327350   {1}{23}       {12}{13}      {12}{13}{23}
%e A327350   {2}{13}       {12}{23}
%e A327350   {3}{12}       {13}{23}
%e A327350   {12}{13}      {12}{13}{23}
%e A327350   {12}{23}
%e A327350   {13}{23}
%e A327350   {1}{2}{3}
%e A327350   {12}{13}{23}
%t A327350 csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A327350 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 A327350 vertConnSys[vts_,eds_]:=Min@@Length/@Select[Subsets[vts],Function[del,Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]];
%t A327350 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],Union@@#==Range[n]&&vertConnSys[Range[n],#]>=k&]],{n,0,4},{k,0,n}]
%Y A327350 Column k = 0 is A307249, or A006126 if empty edges are allowed.
%Y A327350 Column k = 1 is A048143 (clutters), if we assume A048143(0) = A048143(1) = 0.
%Y A327350 Column k = 2 is A275307 (blobs), if we assume A275307(1) = A275307(2) = 0.
%Y A327350 Column k = n - 1 is A327020 (cointersecting antichains).
%Y A327350 The unlabeled version is A327358.
%Y A327350 Negated first differences of rows are A327351.
%Y A327350 BII-numbers of antichains are A326704.
%Y A327350 Antichain covers are A006126.
%Y A327350 Cf. A003465, A014466, A120338, A293606, A293993, A319639, A323818, A327112, A327125, A327334, A327336, A327352, A327356, A327357, A327358.
%K A327350 nonn,tabl,more
%O A327350 0,4
%A A327350 _Gus Wiseman_, Sep 09 2019
%E A327350 a(21) from _Robert Price_, May 24 2021