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A326704
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BII-numbers of antichains of nonempty sets.
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46
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0, 1, 2, 3, 4, 8, 9, 10, 11, 12, 16, 18, 20, 32, 33, 36, 48, 52, 64, 128, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 148, 160, 161, 164, 176, 180, 192, 256, 258, 260, 264, 266, 268, 272, 274, 276, 288, 292, 304, 308, 320, 512, 513, 516, 520, 521, 524
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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, it follows that the BII-number of {{2},{1,3}} is 18.
Elements of a set-system are sometimes called edges. In an antichain of sets, no edge is a subset or superset of any other edge.
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LINKS
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EXAMPLE
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The sequence of all antichains of nonempty sets together with their BII-numbers begins:
0: {}
1: {{1}}
2: {{2}}
3: {{1},{2}}
4: {{1,2}}
8: {{3}}
9: {{1},{3}}
10: {{2},{3}}
11: {{1},{2},{3}}
12: {{1,2},{3}}
16: {{1,3}}
18: {{2},{1,3}}
20: {{1,2},{1,3}}
32: {{2,3}}
33: {{1},{2,3}}
36: {{1,2},{2,3}}
48: {{1,3},{2,3}}
52: {{1,2},{1,3},{2,3}}
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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}];
Select[Range[100], stableQ[bpe/@bpe[#], SubsetQ]&]
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CROSSREFS
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Antichains of sets are counted by A000372.
Antichains of nonempty sets are counted by A014466.
MM-numbers of antichains of multisets are A316476.
BII-numbers of chains of nonempty sets are A326703.
Cf. A000120, A029931, A035327, A048793, A070939, A291166, A302521, A326031, A326675, A326701, A326702.
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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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