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A326880
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BII-numbers of set-systems that are closed under nonempty intersection.
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12
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 38, 39, 40, 41, 42, 43, 46, 47, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 87, 88
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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. 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 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.
The enumeration of these set-systems by number of covered vertices is A326881.
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LINKS
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EXAMPLE
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Most small numbers are in the sequence, but the sequence of non-terms together with the set-systems with those BII-numbers begins:
20: {{1,2},{1,3}}
22: {{2},{1,2},{1,3}}
28: {{1,2},{3},{1,3}}
30: {{2},{1,2},{3},{1,3}}
36: {{1,2},{2,3}}
37: {{1},{1,2},{2,3}}
44: {{1,2},{3},{2,3}}
45: {{1},{1,2},{3},{2,3}}
48: {{1,3},{2,3}}
49: {{1},{1,3},{2,3}}
50: {{2},{1,3},{2,3}}
51: {{1},{2},{1,3},{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}}
84: {{1,2},{1,3},{1,2,3}}
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MATHEMATICA
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bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 100], SubsetQ[bpe/@bpe[#], Intersection@@@Select[Tuples[bpe/@bpe[#], 2], Intersection@@#!={}&]]&]
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CROSSREFS
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Cf. A006126, A048793, A102894, A102895, A102896, A102897, A306445, A326031, A326872, A326874, A326875, A326876, A326881.
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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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