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%I #10 Aug 22 2019 20:41:19
%S 5,6,17,20,24,34,36,40,48,53,54,55,60,61,62,63,65,66,68,71,72,80,86,
%T 87,89,92,93,94,95,96,101,103,106,108,109,110,111,113,114,115,121,122,
%U 123,257,260,272,308,309,310,311,316,317,318,319,320,326,327,342
%N BII-numbers of set-systems with non-spanning edge-connectivity 2.
%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 non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any isolated vertices) to result in a disconnected or empty set-system.
%e The sequence of all set-systems with non-spanning edge-connectivity 2 together with their BII-numbers begins:
%e 5: {{1},{1,2}}
%e 6: {{2},{1,2}}
%e 17: {{1},{1,3}}
%e 20: {{1,2},{1,3}}
%e 24: {{3},{1,3}}
%e 34: {{2},{2,3}}
%e 36: {{1,2},{2,3}}
%e 40: {{3},{2,3}}
%e 48: {{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 65: {{1},{1,2,3}}
%e 66: {{2},{1,2,3}}
%e 68: {{1,2},{1,2,3}}
%e 71: {{1},{2},{1,2},{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 edgeConn[y_]:=If[Length[csm[bpe/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[bpe/@#]]!=1&]];
%t Select[Range[0,100],edgeConn[bpe[#]]==2&]
%Y Positions of 2's in A326787.
%Y BII-numbers for vertex-connectivity 2 are A327082.
%Y BII-numbers for non-spanning edge-connectivity 1 are A327099.
%Y BII-numbers for non-spanning edge-connectivity > 1 are A327102.
%Y BII-numbers for spanning edge-connectivity 2 are A327108.
%Y Cf. A007146, A048793, A052446, A059166, A070939, A095983, A263296, A322335, A322338, A322395, A326031, A327041, A327069, A327111.
%K nonn
%O 1,1
%A _Gus Wiseman_, Aug 20 2019
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