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A327352 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.

%I A327352 #5 Sep 11 2019 20:21:47
%S A327352 1,1,1,4,1,14,4,1,83,59,23,2,1232,2551,2792,887,107,10,1
%N A327352 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 A327352 An antichain is a set of sets, none of which is a subset of any other.
%C A327352 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 A327352 Triangle begins:
%e A327352      1
%e A327352      1    1
%e A327352      4    1
%e A327352     14    4    1
%e A327352     83   59   23    2
%e A327352   1232 2551 2792  887  107   10    1
%e A327352 Row n = 3 counts the following antichains:
%e A327352   {}             {{1,2,3}}      {{1,2},{1,3},{2,3}}
%e A327352   {{1}}          {{1,2},{1,3}}
%e A327352   {{2}}          {{1,2},{2,3}}
%e A327352   {{3}}          {{1,3},{2,3}}
%e A327352   {{1,2}}
%e A327352   {{1,3}}
%e A327352   {{2,3}}
%e A327352   {{1},{2}}
%e A327352   {{1},{3}}
%e A327352   {{2},{3}}
%e A327352   {{1},{2,3}}
%e A327352   {{2},{1,3}}
%e A327352   {{3},{1,2}}
%e A327352   {{1},{2},{3}}
%t A327352 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 A327352 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 A327352 spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
%t A327352 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 A327352 Row sums are A014466.
%Y A327352 Column k = 0 is A327355.
%Y A327352 The unlabeled version is A327438.
%Y A327352 Cf. A052446, A327062, A327071, A327103, A327111, A327144, A327351, A327353.
%K A327352 nonn,tabf,more
%O A327352 0,4
%A A327352 _Gus Wiseman_, Sep 10 2019