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Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of nonempty subsets of {1..n} with spanning edge-connectivity k.
8

%I #5 Sep 11 2019 20:21:47

%S 1,1,1,4,1,14,4,1,83,59,23,2,1232,2551,2792,887,107,10,1

%N Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of nonempty subsets of {1..n} with spanning edge-connectivity k.

%C An antichain is a set of sets, none of which is a subset of any other.

%C The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices.

%e Triangle begins:

%e 1

%e 1 1

%e 4 1

%e 14 4 1

%e 83 59 23 2

%e 1232 2551 2792 887 107 10 1

%e Row n = 3 counts the following antichains:

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

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

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

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

%e {{1,2}}

%e {{1,3}}

%e {{2,3}}

%e {{1},{2}}

%e {{1},{3}}

%e {{2},{3}}

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

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

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

%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 spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];

%t Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],spanEdgeConn[Range[n],#]==k&]],{n,0,4},{k,0,2^n}]//.{foe___,0}:>{foe}

%Y Row sums are A014466.

%Y Column k = 0 is A327355.

%Y The unlabeled version is A327438.

%Y Cf. A052446, A327062, A327071, A327103, A327111, A327144, A327351, A327353.

%K nonn,tabf,more

%O 0,4

%A _Gus Wiseman_, Sep 10 2019