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 A327101 BII-numbers of 2-cut-connected set-systems (cut-connectivity >= 2). 10
 4, 5, 6, 7, 16, 17, 24, 25, 32, 34, 40, 42, 52, 53, 54, 55, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107 (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. A set-system is 2-cut-connected if any single vertex can be removed (along with any empty edges) without making the set-system disconnected or empty. Except for cointersecting set-systems (A326853), this is the same as 2-vertex-connectivity. LINKS FORMULA If (*) is intersection and (-) is complement, we have A327101 * A326704 = A326751 - A058891, i.e., the intersection of A327101 (this sequence) with A326704 (antichains) is the complement of A058891 (singletons) in A326751 (blobs). EXAMPLE The sequence of all 2-cut-connected set-systems together with their BII-numbers begins:    4: {{1,2}}    5: {{1},{1,2}}    6: {{2},{1,2}}    7: {{1},{2},{1,2}}   16: {{1,3}}   17: {{1},{1,3}}   24: {{3},{1,3}}   25: {{1},{3},{1,3}}   32: {{2,3}}   34: {{2},{2,3}}   40: {{3},{2,3}}   42: {{2},{3},{2,3}}   52: {{1,2},{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}} 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]]]]]]]]]; cutConnSys[vts_, eds_]:=If[Length[vts]==1, 1, Min@@Length/@Select[Subsets[vts], Function[del, csm[DeleteCases[DeleteCases[eds, Alternatives@@del, {2}], {}]]!={Complement[vts, del]}]]]; Select[Range[0, 100], cutConnSys[Union@@bpe/@bpe[#], bpe/@bpe[#]]>=2&] CROSSREFS Positions of numbers >= 2 in A326786. 2-cut-connected graphs are counted by A013922, if we assume A013922(2) = 0. 2-cut-connected integer partitions are counted by A322387. BII-numbers for cut-connectivity 2 are A327082. BII-numbers for cut-connectivity 1 are A327098. BII-numbers for non-spanning edge-connectivity >= 2 are A327102. BII-numbers for spanning edge-connectivity >= 2 are A327109. Covering 2-cut-connected set-systems are counted by A327112. Covering set-systems with cut-connectivity 2 are counted by A327113. The labeled cut-connectivity triangle is A327125, with unlabeled version A327127. Cf. A000120, A002218, A048793, A070939, A259862, A322388, A326031, A326749, A327097, A327108. Sequence in context: A294228 A046300 A053738 * A327082 A154787 A233035 Adjacent sequences:  A327098 A327099 A327100 * A327102 A327103 A327104 KEYWORD nonn AUTHOR Gus Wiseman, Aug 22 2019 STATUS approved

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Last modified April 9 10:32 EDT 2020. Contains 333348 sequences. (Running on oeis4.)