|
%I #10 Jul 27 2019 14:57:51
%S 0,1,2,4,8,16,32,52,64,128,256,512,772,816,820,832,1024,1072,1088,
%T 2048,2320,2340,2356,2368,2580,2592,2612,2624,2836,2852,2864,2868,
%U 2880,3088,3104,3120,3136,4096,4132,4160,4612,4640,4644,4672,5120,5152,5184,8192
%N BII-numbers of blobs.
%C A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every 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.
%C Elements of a set-system are sometimes called edges. In an antichain, no edge is a subset or superset of any other edge. In a 2-vertex-connected set-system, at least two vertices must be removed to make the set-system disconnected. A blob is a connected, 2-vertex-connected antichain of finite, nonempty sets, or, equivalently, a 2-vertex-connected clutter.
%H Gus Wiseman, <a href="http://www.mathematica-journal.com/2017/12/every-clutter-is-a-tree-of-blobs/">Every Clutter Is a Tree of Blobs</a>, The Mathematica Journal, Vol. 19, 2017.
%e The sequence of all blobs together with their BII-numbers begins:
%e 0: {}
%e 1: {{1}}
%e 2: {{2}}
%e 4: {{1,2}}
%e 8: {{3}}
%e 16: {{1,3}}
%e 32: {{2,3}}
%e 52: {{1,2},{1,3},{2,3}}
%e 64: {{1,2,3}}
%e 128: {{4}}
%e 256: {{1,4}}
%e 512: {{2,4}}
%e 772: {{1,2},{1,4},{2,4}}
%e 816: {{1,3},{2,3},{1,4},{2,4}}
%e 820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
%e 832: {{1,2,3},{1,4},{2,4}}
%e 1024: {{1,2,4}}
%e 1072: {{1,3},{2,3},{1,2,4}}
%e 1088: {{1,2,3},{1,2,4}}
%e 2048: {{3,4}}
%e 2320: {{1,3},{1,4},{3,4}}
%e 2340: {{1,2},{2,3},{1,4},{3,4}}
%e 2356: {{1,2},{1,3},{2,3},{1,4},{3,4}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t tvcQ[eds_]:=And@@Table[Length[csm[DeleteCases[eds,i,{2}]]]<=1,{i,Union@@eds}];
%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 Select[Range[0,1000],stableQ[bpe/@bpe[#],SubsetQ]&&Length[csm[bpe/@bpe[#]]]<=1&&tvcQ[bpe/@bpe[#]]&]
%Y Cf. A000120, A002218, A013922 (2-vertex-connected graphs), A030019, A048143 (clutters), A048793, A070939, A095983, A275307 (spanning blobs), A304118, A304887, A322117, A322397 (2-edge-connected clutters), A326031.
%Y Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326752 (hypertrees), A326754 (covers).
%K nonn
%O 1,3
%A _Gus Wiseman_, Jul 23 2019
|
