cp's OEIS Frontend

A327101 BII-numbers of 2-cut-connected set-systems (cut-connectivity >= 2).

%I A327101 #10 Sep 01 2019 22:03:25
%S A327101 4,5,6,7,16,17,24,25,32,34,40,42,52,53,54,55,60,61,62,63,64,65,66,67,
%T A327101 68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,
%U A327101 91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107
%N A327101 BII-numbers of 2-cut-connected set-systems (cut-connectivity >= 2).
%C A327101 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 A327101 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.
%F A327101 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).
%e A327101 The sequence of all 2-cut-connected set-systems together with their BII-numbers begins:
%e A327101    4: {{1,2}}
%e A327101    5: {{1},{1,2}}
%e A327101    6: {{2},{1,2}}
%e A327101    7: {{1},{2},{1,2}}
%e A327101   16: {{1,3}}
%e A327101   17: {{1},{1,3}}
%e A327101   24: {{3},{1,3}}
%e A327101   25: {{1},{3},{1,3}}
%e A327101   32: {{2,3}}
%e A327101   34: {{2},{2,3}}
%e A327101   40: {{3},{2,3}}
%e A327101   42: {{2},{3},{2,3}}
%e A327101   52: {{1,2},{1,3},{2,3}}
%e A327101   53: {{1},{1,2},{1,3},{2,3}}
%e A327101   54: {{2},{1,2},{1,3},{2,3}}
%e A327101   55: {{1},{2},{1,2},{1,3},{2,3}}
%e A327101   60: {{1,2},{3},{1,3},{2,3}}
%e A327101   61: {{1},{1,2},{3},{1,3},{2,3}}
%e A327101   62: {{2},{1,2},{3},{1,3},{2,3}}
%e A327101   63: {{1},{2},{1,2},{3},{1,3},{2,3}}
%t A327101 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A327101 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 A327101 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]}]]];
%t A327101 Select[Range[0,100],cutConnSys[Union@@bpe/@bpe[#],bpe/@bpe[#]]>=2&]
%Y A327101 Positions of numbers >= 2 in A326786.
%Y A327101 2-cut-connected graphs are counted by A013922, if we assume A013922(2) = 0.
%Y A327101 2-cut-connected integer partitions are counted by A322387.
%Y A327101 BII-numbers for cut-connectivity 2 are A327082.
%Y A327101 BII-numbers for cut-connectivity 1 are A327098.
%Y A327101 BII-numbers for non-spanning edge-connectivity >= 2 are A327102.
%Y A327101 BII-numbers for spanning edge-connectivity >= 2 are A327109.
%Y A327101 Covering 2-cut-connected set-systems are counted by A327112.
%Y A327101 Covering set-systems with cut-connectivity 2 are counted by A327113.
%Y A327101 The labeled cut-connectivity triangle is A327125, with unlabeled version A327127.
%Y A327101 Cf. A000120, A002218, A048793, A070939, A259862, A322388, A326031, A326749, A327097, A327108.
%K A327101 nonn
%O A327101 1,1
%A A327101 _Gus Wiseman_, Aug 22 2019