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A326751
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BII-numbers of blobs.
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15
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0, 1, 2, 4, 8, 16, 32, 52, 64, 128, 256, 512, 772, 816, 820, 832, 1024, 1072, 1088, 2048, 2320, 2340, 2356, 2368, 2580, 2592, 2612, 2624, 2836, 2852, 2864, 2868, 2880, 3088, 3104, 3120, 3136, 4096, 4132, 4160, 4612, 4640, 4644, 4672, 5120, 5152, 5184, 8192
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OFFSET
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1,3
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COMMENTS
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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.
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.
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LINKS
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EXAMPLE
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The sequence of all blobs together with their BII-numbers begins:
0: {}
1: {{1}}
2: {{2}}
4: {{1,2}}
8: {{3}}
16: {{1,3}}
32: {{2,3}}
52: {{1,2},{1,3},{2,3}}
64: {{1,2,3}}
128: {{4}}
256: {{1,4}}
512: {{2,4}}
772: {{1,2},{1,4},{2,4}}
816: {{1,3},{2,3},{1,4},{2,4}}
820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
832: {{1,2,3},{1,4},{2,4}}
1024: {{1,2,4}}
1072: {{1,3},{2,3},{1,2,4}}
1088: {{1,2,3},{1,2,4}}
2048: {{3,4}}
2320: {{1,3},{1,4},{3,4}}
2340: {{1,2},{2,3},{1,4},{3,4}}
2356: {{1,2},{1,3},{2,3},{1,4},{3,4}}
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MATHEMATICA
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bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
tvcQ[eds_]:=And@@Table[Length[csm[DeleteCases[eds, i, {2}]]]<=1, {i, Union@@eds}];
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]]]]]]]]];
Select[Range[0, 1000], stableQ[bpe/@bpe[#], SubsetQ]&&Length[csm[bpe/@bpe[#]]]<=1&&tvcQ[bpe/@bpe[#]]&]
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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