login
A327016
BII-numbers of finite T_0 topologies without their empty set.
2
0, 1, 2, 5, 6, 7, 8, 17, 24, 25, 34, 40, 42, 69, 70, 71, 81, 85, 87, 88, 89, 93, 98, 102, 103, 104, 106, 110, 120, 121, 122, 127, 128, 257, 384, 385, 514, 640, 642, 1029, 1030, 1031, 1281, 1285, 1287, 1408, 1409, 1413, 1538, 1542, 1543, 1664, 1666, 1670, 1920
OFFSET
1,3
COMMENTS
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_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.
EXAMPLE
The sequence of all finite T_0 topologies without their empty set together with their BII-numbers begins:
0: {}
1: {{1}}
2: {{2}}
5: {{1},{1,2}}
6: {{2},{1,2}}
7: {{1},{2},{1,2}}
8: {{3}}
17: {{1},{1,3}}
24: {{3},{1,3}}
25: {{1},{3},{1,3}}
34: {{2},{2,3}}
40: {{3},{2,3}}
42: {{2},{3},{2,3}}
69: {{1},{1,2},{1,2,3}}
70: {{2},{1,2},{1,2,3}}
71: {{1},{2},{1,2},{1,2,3}}
81: {{1},{1,3},{1,2,3}}
85: {{1},{1,2},{1,3},{1,2,3}}
87: {{1},{2},{1,2},{1,3},{1,2,3}}
88: {{3},{1,3},{1,2,3}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 1000], UnsameQ@@dual[bpe/@bpe[#]]&&SubsetQ[bpe/@bpe[#], Union[Union@@@Tuples[bpe/@bpe[#], 2], DeleteCases[Intersection@@@Tuples[bpe/@bpe[#], 2], {}]]]&]
CROSSREFS
T_0 topologies are A001035, with unlabeled version A000112.
BII-numbers of topologies without their empty set are A326876.
BII-numbers of T_0 set-systems are A326947.
Sequence in context: A246263 A191205 A043047 * A271107 A014355 A048972
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 14 2019
STATUS
approved