# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a327353 Showing 1-1 of 1 %I A327353 #7 Sep 11 2019 20:21:53 %S A327353 1,1,1,2,3,8,7,3,1,53,27,45,36,6,747,511,1497,2085,1540,693,316,135, %T A327353 45,10,1 %N A327353 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of subsets of {1..n} with non-spanning edge-connectivity k. %C A327353 An antichain is a set of sets, none of which is a subset of any other. %C A327353 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 A327353 Triangle begins: %e A327353 1 %e A327353 1 1 %e A327353 2 3 %e A327353 8 7 3 1 %e A327353 53 27 45 36 6 %e A327353 747 511 1497 2085 1540 693 316 135 45 10 1 %e A327353 Row n = 3 counts the following antichains: %e A327353 {} {{1}} {{1,2},{1,3}} {{1,2},{1,3},{2,3}} %e A327353 {{1},{2}} {{2}} {{1,2},{2,3}} %e A327353 {{1},{3}} {{3}} {{1,3},{2,3}} %e A327353 {{2},{3}} {{1,2}} %e A327353 {{1},{2,3}} {{1,3}} %e A327353 {{2},{1,3}} {{2,3}} %e A327353 {{3},{1,2}} {{1,2,3}} %e A327353 {{1},{2},{3}} %t A327353 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 A327353 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 A327353 eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]]; %t A327353 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],eConn[#]==k&]],{n,0,4},{k,0,2^n}]//.{foe___,0}:>{foe} %Y A327353 Row sums are A014466. %Y A327353 Column k = 0 is A327354. %Y A327353 The covering case is A327357. %Y A327353 The version for spanning edge-connectivity is A327352. %Y A327353 The specialization to simple graphs is A327148, with covering case A327149, unlabeled version A327236, and unlabeled covering case A327201. %Y A327353 Cf. A052446, A307249, A326704, A326787, A327071, A327351, A327355. %K A327353 nonn,tabf,more %O A327353 0,4 %A A327353 _Gus Wiseman_, Sep 10 2019 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE