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 A327097 BII-numbers of set-systems with non-spanning edge-connectivity 2. 13
 5, 6, 17, 20, 24, 34, 36, 40, 48, 53, 54, 55, 60, 61, 62, 63, 65, 66, 68, 71, 72, 80, 86, 87, 89, 92, 93, 94, 95, 96, 101, 103, 106, 108, 109, 110, 111, 113, 114, 115, 121, 122, 123, 257, 260, 272, 308, 309, 310, 311, 316, 317, 318, 319, 320, 326, 327, 342 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. 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. LINKS EXAMPLE The sequence of all set-systems with non-spanning edge-connectivity 2 together with their BII-numbers begins:    5: {{1},{1,2}}    6: {{2},{1,2}}   17: {{1},{1,3}}   20: {{1,2},{1,3}}   24: {{3},{1,3}}   34: {{2},{2,3}}   36: {{1,2},{2,3}}   40: {{3},{2,3}}   48: {{1,3},{2,3}}   53: {{1},{1,2},{1,3},{2,3}}   54: {{2},{1,2},{1,3},{2,3}}   55: {{1},{2},{1,2},{1,3},{2,3}}   60: {{1,2},{3},{1,3},{2,3}}   61: {{1},{1,2},{3},{1,3},{2,3}}   62: {{2},{1,2},{3},{1,3},{2,3}}   63: {{1},{2},{1,2},{3},{1,3},{2,3}}   65: {{1},{1,2,3}}   66: {{2},{1,2,3}}   68: {{1,2},{1,2,3}}   71: {{1},{2},{1,2},{1,2,3}} MATHEMATICA bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1]; 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]]]]]]]]]; edgeConn[y_]:=If[Length[csm[bpe/@y]]!=1, 0, Length[y]-Max@@Length/@Select[Union[Subsets[y]], Length[csm[bpe/@#]]!=1&]]; Select[Range[0, 100], edgeConn[bpe[#]]==2&] CROSSREFS Positions of 2's in A326787. BII-numbers for vertex-connectivity 2 are A327082. BII-numbers for non-spanning edge-connectivity 1 are A327099. BII-numbers for non-spanning edge-connectivity > 1 are A327102. BII-numbers for spanning edge-connectivity 2 are A327108. Cf. A007146, A048793, A052446, A059166, A070939, A095983, A263296, A322335, A322338, A322395, A326031, A327041, A327069, A327111. Sequence in context: A059013 A327102 A191144 * A035595 A099571 A041997 Adjacent sequences:  A327094 A327095 A327096 * A327098 A327099 A327100 KEYWORD nonn AUTHOR Gus Wiseman, Aug 20 2019 STATUS approved

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Last modified April 21 14:50 EDT 2021. Contains 343154 sequences. (Running on oeis4.)