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%I
%S 52,53,54,55,60,61,62,63,84,85,86,87,92,93,94,95,100,101,102,103,108,
%T 109,110,111,112,113,114,115,120,121,122,123,772,773,774,775,816,817,
%U 818,819,820,821,822,823,824,825,826,827,828,829,830,831,848,849,850
%N BII-numbers of set-systems with spanning edge-connectivity 2.
%C Differs from A327109 in lacking 116, 117, 118, 119, 124, 125, 126, 127, ...
%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 disconnected or empty set-system.
%e The sequence of all set-systems with spanning edge-connectivity 2 together with their BII-numbers begins:
%e 52: {{1,2},{1,3},{2,3}}
%e 53: {{1},{1,2},{1,3},{2,3}}
%e 54: {{2},{1,2},{1,3},{2,3}}
%e 55: {{1},{2},{1,2},{1,3},{2,3}}
%e 60: {{1,2},{3},{1,3},{2,3}}
%e 61: {{1},{1,2},{3},{1,3},{2,3}}
%e 62: {{2},{1,2},{3},{1,3},{2,3}}
%e 63: {{1},{2},{1,2},{3},{1,3},{2,3}}
%e 84: {{1,2},{1,3},{1,2,3}}
%e 85: {{1},{1,2},{1,3},{1,2,3}}
%e 86: {{2},{1,2},{1,3},{1,2,3}}
%e 87: {{1},{2},{1,2},{1,3},{1,2,3}}
%e 92: {{1,2},{3},{1,3},{1,2,3}}
%e 93: {{1},{1,2},{3},{1,3},{1,2,3}}
%e 94: {{2},{1,2},{3},{1,3},{1,2,3}}
%e 95: {{1},{2},{1,2},{3},{1,3},{1,2,3}}
%e 100: {{1,2},{2,3},{1,2,3}}
%e 101: {{1},{1,2},{2,3},{1,2,3}}
%e 102: {{2},{1,2},{2,3},{1,2,3}}
%e 103: {{1},{2},{1,2},{2,3},{1,2,3}}
%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 Select[Range[0,100],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]==2&]
%Y Positions of 2's in A327144.
%Y Graphs with spanning edge-connectivity >= 2 are counted by A095983.
%Y Graphs with spanning edge-connectivity 2 are counted by A327146.
%Y Set-systems with spanning edge-connectivity 2 are counted by A327130.
%Y BII-numbers for non-spanning edge-connectivity 2 are A327097.
%Y BII-numbers for spanning edge-connectivity >= 2 are A327109.
%Y BII-numbers for spanning edge-connectivity 1 are A327111.
%Y Cf. A326749, A326787, A327041, A327069, A327075, A327102, A327103, A327104.
%K nonn
%O 1,1
%A _Gus Wiseman_, Aug 23 2019
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