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%I #6 Sep 01 2019 08:41:53
%S 0,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,1,1,1,1,1,0,0,1,1,1,1,1,0,
%T 1,0,1,1,1,1,1,0,1,0,1,1,1,1,1,1,1,1,2,2,2,2,1,1,1,1,2,2,2,2,1,1,1,1,
%U 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2
%N 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 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 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 Positions of first appearances of each integer together with the corresponding set-systems:
%e 0: {}
%e 1: {{1}}
%e 52: {{1,2},{1,3},{2,3}}
%e 116: {{1,2},{1,3},{2,3},{1,2,3}}
%e 3952: {{1,3},{2,3},{1,4},{2,4},{3,4},{1,2,3},{1,2,4}}
%e 8052: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{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[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 spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
%t Table[spanEdgeConn[Union@@bpe/@bpe[n],bpe/@bpe[n]],{n,0,100}]
%Y Dominated by A327103.
%Y The same for cut-connectivity is A326786.
%Y The same for non-spanning edge-connectivity is A326787.
%Y The same for vertex-connectivity is A327051.
%Y Positions of 1's are A327111.
%Y Positions of 2's are A327108.
%Y Positions of first appearance of each integer are A327147.
%Y Cf. A000120, A048793, A070939, A322338, A323818, A326031, A327041, A327069, A327076, A327130, A327145.
%K nonn
%O 0,53
%A _Gus Wiseman_, Aug 31 2019
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