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%I #10 Sep 02 2019 02:17:25
%S 0,1,1,0,2,2,2,2,1,0,0,0,0,0,0,0,2,2,0,0,1,1,1,1,2,2,0,0,1,1,1,1,2,0,
%T 2,0,1,1,1,1,2,0,2,0,1,1,1,1,1,1,1,1,3,3,3,3,1,1,1,1,3,3,3,3,3,3,3,3,
%U 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3
%N Cut-connectivity of the set-system with BII-number n.
%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 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.
%C Elements of a set-system are sometimes called edges. The cut-connectivity of a set-system is the minimum number of vertices that must be removed (together with any resulting empty or duplicate edges) to obtain a disconnected or empty set-system. Except for cointersecting set-systems (A326853), this is the same as vertex-connectivity (A327051).
%e Positions of first appearances of each integer, together with the corresponding set-systems, are:
%e 0: {}
%e 1: {{1}}
%e 4: {{1,2}}
%e 52: {{1,2},{1,3},{2,3}}
%e 2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
%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 vertConn[y_]:=If[Length[csm[bpe/@y]]!=1,0,Min@@Length/@Select[Subsets[Union@@bpe/@y],Function[del,Length[csm[DeleteCases[DeleteCases[bpe/@y,Alternatives@@del,{2}],{}]]]!=1]]];
%t Table[vertConn[bpe[n]],{n,0,100}]
%Y Cf. A000120, A013922, A029931, A048793, A070939, A305078, A322388, A322389 (same for MM-numbers), A322390, A326031, A326701, A326749, A326753, A326787 (edge-connectivity), A327051 (vertex-connectivity).
%K nonn
%O 0,5
%A _Gus Wiseman_, Jul 25 2019
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