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%I #10 Sep 26 2019 03:43:10
%S 0,1,1,0,1,2,2,1,1,0,0,0,0,0,0,0,1,2,0,0,2,3,1,1,2,1,0,0,1,1,1,1,1,0,
%T 2,0,2,1,3,1,2,0,1,0,1,1,1,1,2,1,1,1,3,2,2,2,3,1,1,1,2,2,2,2,1,2,2,1,
%U 2,3,3,2,2,1,1,1,1,1,1,1,2,3,1,1,3,4,2
%N Non-spanning edge-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 non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed to obtain a graph whose edge-set is disconnected or empty.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/K-edge-connected_graph">k-edge-connected graph</a>
%e Positions of first appearances of each integer together with the corresponding set-systems:
%e 0: {}
%e 1: {{1}}
%e 5: {{1},{1,2}}
%e 21: {{1},{1,2},{1,3}}
%e 85: {{1},{1,2},{1,3},{1,2,3}}
%e 341: {{1},{1,2},{1,3},{1,4},{1,2,3}}
%e 1365: {{1},{1,2},{1,3},{1,4},{1,2,3},{1,2,4}}
%e 5461: {{1},{1,2},{1,3},{1,4},{1,2,3},{1,2,4},{1,3,4}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%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 eConn[sys_]:=Length[sys]-Max@@Length/@Select[Subsets[sys],Length[csm[#]]!=1&];
%t Table[eConn[bpe/@bpe[n]],{n,0,100}]
%Y Cf. A000120, A013922, A048793, A070939, A095983, A322336, A322338 (same for MM-numbers), A326031, A326749, A326753, A326786 (vertex-connectivity).
%K nonn
%O 0,6
%A _Gus Wiseman_, Jul 25 2019
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