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%I #14 Sep 01 2019 22:03:17
%S 1,2,8,20,36,48,128,260,272,276,292,304,308,320,516,532,544,548,560,
%T 564,576,768,784,788,800,804,1040,1056,2064,2068,2080,2084,2096,2100,
%U 2112,2304,2308,2324,2336,2352,2560,2564,2576,2596,2608,2816,2820,2832,2848
%N BII-numbers of antichains of sets with cut-connectivity 1.
%C A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.
%C We define the cut-connectivity of a set-system to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty set-system, with the exception that a set-system with one vertex has cut-connectivity 1. Except for cointersecting set-systems (A326853, A327039, A327040), this is the same as vertex-connectivity (A327334, A327051).
%F If (+) is union and (-) is complement, we have A327100 = A058891 + (A326750 - A326751).
%e The sequence of all antichains of sets with vertex-connectivity 1 together with their BII-numbers begins:
%e 1: {{1}}
%e 2: {{2}}
%e 8: {{3}}
%e 20: {{1,2},{1,3}}
%e 36: {{1,2},{2,3}}
%e 48: {{1,3},{2,3}}
%e 128: {{4}}
%e 260: {{1,2},{1,4}}
%e 272: {{1,3},{1,4}}
%e 276: {{1,2},{1,3},{1,4}}
%e 292: {{1,2},{2,3},{1,4}}
%e 304: {{1,3},{2,3},{1,4}}
%e 308: {{1,2},{1,3},{2,3},{1,4}}
%e 320: {{1,2,3},{1,4}}
%e 516: {{1,2},{2,4}}
%e 532: {{1,2},{1,3},{2,4}}
%e 544: {{2,3},{2,4}}
%e 548: {{1,2},{2,3},{2,4}}
%e 560: {{1,3},{2,3},{2,4}}
%e 564: {{1,2},{1,3},{2,3},{2,4}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t cutConnSys[vts_,eds_]:=If[Length[vts]==1,1,Min@@Length/@Select[Subsets[vts],Function[del,csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]]];
%t Select[Range[0,100],stableQ[bpe/@bpe[#],SubsetQ]&&cutConnSys[Union@@bpe/@bpe[#],bpe/@bpe[#]]==1&]
%Y Positions of 1's in A326786.
%Y The graphical case is A327114.
%Y BII numbers of antichains with vertex-connectivity >= 1 are A326750.
%Y BII-numbers for cut-connectivity 2 are A327082.
%Y BII-numbers for cut-connectivity 1 are A327098.
%Y Cf. A000120, A000372, A006126, A048143, A048793, A070939, A322390, A326031, A326749, A326751, A327071, A327111.
%K nonn
%O 1,2
%A _Gus Wiseman_, Aug 22 2019
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