A327350 - OEIS

%I #14 May 25 2021 01:40:58

%S 1,1,0,2,1,0,9,5,2,0,114,84,44,17,0,6894,6348,4983,3141,1451,0,7785062

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

%C 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 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 If empty edges are allowed, we have T(0,0) = 2.

%e Triangle begins:

%e      1

%e      1    0

%e      2    1    0

%e      9    5    2    0

%e    114   84   44   17    0

%e   6894 6348 4983 3141 1451    0

%e The antichains counted in row n = 3:

%e   {123}         {123}         {123}

%e   {1}{23}       {12}{13}      {12}{13}{23}

%e   {2}{13}       {12}{23}

%e   {3}{12}       {13}{23}

%e   {12}{13}      {12}{13}{23}

%e   {12}{23}

%e   {13}{23}

%e   {1}{2}{3}

%e   {12}{13}{23}

%t 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 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 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 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],Union@@#==Range[n]&&vertConnSys[Range[n],#]>=k&]],{n,0,4},{k,0,n}]

%Y Column k = 0 is A307249, or A006126 if empty edges are allowed.

%Y Column k = 1 is A048143 (clutters), if we assume A048143(0) = A048143(1) = 0.

%Y Column k = 2 is A275307 (blobs), if we assume A275307(1) = A275307(2) = 0.

%Y Column k = n - 1 is A327020 (cointersecting antichains).

%Y The unlabeled version is A327358.

%Y Negated first differences of rows are A327351.

%Y BII-numbers of antichains are A326704.

%Y Antichain covers are A006126.

%Y Cf. A003465, A014466, A120338, A293606, A293993, A319639, A323818, A327112, A327125, A327334, A327336, A327352, A327356, A327357, A327358.

%K nonn,tabl,more

%O 0,4

%A _Gus Wiseman_, Sep 09 2019

%E a(21) from _Robert Price_, May 24 2021