%I #4 Sep 11 2019 20:22:21

%S 1,0,1,1,1,4,1,3,1,30,13,33,32,6,546,421,1302,1915,1510,693,316,135,

%T 45,10,1

%N Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of sets covering n vertices with non-spanning edge-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 non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty set-system.

%e Triangle begins:

%e 1

%e 0 1

%e 1 1

%e 4 1 3 1

%e 30 13 33 32 6

%e 546 421 1302 1915 1510 693 316 135 45 10 1

%e Row n = 3 counts the following antichains:

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

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

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

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

%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 eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];

%t Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],Union@@#==Range[n]&&eConn[#]==k&]],{n,0,5},{k,0,2^n}]//.{foe___,0}:>{foe}

%Y Row sums are A307249.

%Y Column k = 0 is A120338.

%Y The non-covering version is A327353.

%Y The version for spanning edge-connectivity is A327352.

%Y The specialization to simple graphs is A327149, with unlabeled version A327201.

%Y Cf. A014466, A293606, A326704, A326787, A327071, A327148, A327236, A327351.

%K nonn,tabf,more

%O 0,6

%A _Gus Wiseman_, Sep 11 2019