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A326947
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BII-numbers of T_0 set-systems.
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28
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0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 69, 70, 71, 73, 74, 75, 77, 78
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OFFSET
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1,3
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COMMENTS
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The dual of a set-system has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).
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. Elements of a set-system are sometimes called edges.
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LINKS
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EXAMPLE
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The sequence of all T_0 set-systems together with their BII numbers begins:
0: {}
1: {{1}}
2: {{2}}
3: {{1},{2}}
5: {{1},{1,2}}
6: {{2},{1,2}}
7: {{1},{2},{1,2}}
8: {{3}}
9: {{1},{3}}
10: {{2},{3}}
11: {{1},{2},{3}}
13: {{1},{1,2},{3}}
14: {{2},{1,2},{3}}
15: {{1},{2},{1,2},{3}}
17: {{1},{1,3}}
19: {{1},{2},{1,3}}
20: {{1,2},{1,3}}
21: {{1},{1,2},{1,3}}
22: {{2},{1,2},{1,3}}
23: {{1},{2},{1,2},{1,3}}
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MATHEMATICA
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bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
TZQ[sys_]:=UnsameQ@@dual[sys];
Select[Range[0, 100], TZQ[bpe/@bpe[#]]&]
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CROSSREFS
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T_0 set-systems are counted by A326940, with unlabeled version A326946.
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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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