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A327351 Triangle read by rows where T(n,k) is the number of antichains of nonempty sets covering n vertices with vertex-connectivity exactly k.

%I A327351 #12 May 28 2021 11:03:22
%S A327351 1,1,0,1,1,0,4,3,2,0,30,40,27,17,0,546,1365,1842,1690,1451,0,41334
%N A327351 Triangle read by rows where T(n,k) is the number of antichains of nonempty sets covering n vertices with vertex-connectivity exactly k.
%C A327351 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 A327351 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 A327351 If empty edges are allowed, we have T(0,0) = 2.
%e A327351 Triangle begins:
%e A327351     1
%e A327351     1    0
%e A327351     1    1    0
%e A327351     4    3    2    0
%e A327351    30   40   27   17    0
%e A327351   546 1365 1842 1690 1451    0
%t A327351 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 A327351 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 A327351 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 A327351 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],Union@@#==Range[n]&&vertConnSys[Range[n],#]==k&]],{n,0,4},{k,0,n}]
%Y A327351 Row sums are A307249, or A006126 if empty edges are allowed.
%Y A327351 Column k = 0 is A120338, if we assume A120338(0) = A120338(1) = 1.
%Y A327351 Column k = 1 is A327356.
%Y A327351 Column k = n - 1 is A327020.
%Y A327351 The unlabeled version is A327359.
%Y A327351 The version for vertex-connectivity >= k is A327350.
%Y A327351 The version for spanning edge-connectivity is A327352.
%Y A327351 The version for non-spanning edge-connectivity is A327353, with covering case A327357.
%Y A327351 Cf. A003465, A006126, A014466, A048143, A293993, A323818, A326704, A327125, A327334, A327336.
%K A327351 nonn,tabl,more
%O A327351 0,7
%A A327351 _Gus Wiseman_, Sep 09 2019
%E A327351 a(21) from _Robert Price_, May 28 2021