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A326979
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BII-numbers of T_1 set-systems.
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17
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0, 1, 2, 3, 7, 8, 9, 10, 11, 15, 25, 27, 30, 31, 42, 43, 45, 47, 51, 52, 53, 54, 55, 59, 60, 61, 62, 63, 75, 79, 91, 94, 95, 107, 109, 111, 115, 116, 117, 118, 119, 123, 124, 125, 126, 127, 128, 129, 130, 131, 135, 136, 137, 138, 139, 143, 153, 155, 158, 159
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OFFSET
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1,3
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COMMENTS
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A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_1 condition means that the dual is a (strict) antichain, meaning that none of its edges is a subset of any other.
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_1 set-systems together with their BII-numbers begins:
0: {}
1: {{1}}
2: {{2}}
3: {{1},{2}}
7: {{1},{2},{1,2}}
8: {{3}}
9: {{1},{3}}
10: {{2},{3}}
11: {{1},{2},{3}}
15: {{1},{2},{1,2},{3}}
25: {{1},{3},{1,3}}
27: {{1},{2},{3},{1,3}}
30: {{2},{1,2},{3},{1,3}}
31: {{1},{2},{1,2},{3},{1,3}}
42: {{2},{3},{2,3}}
43: {{1},{2},{3},{2,3}}
45: {{1},{1,2},{3},{2,3}}
47: {{1},{2},{1,2},{3},{2,3}}
51: {{1},{2},{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];
dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Select[Range[0, 100], UnsameQ@@dual[bpe/@bpe[#]]&&stableQ[dual[bpe/@bpe[#]], SubsetQ]&]
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CROSSREFS
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BII-numbers of T_0 set-systems are A326947.
BII-numbers of set-systems whose dual is a weak antichain are A326966.
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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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