A326876 BII-numbers of finite topologies without their empty set.

0, 1, 2, 4, 5, 6, 7, 8, 16, 17, 24, 25, 32, 34, 40, 42, 64, 65, 66, 68, 69, 70, 71, 72, 76, 80, 81, 82, 85, 87, 88, 89, 93, 96, 97, 98, 102, 103, 104, 106, 110, 120, 121, 122, 127, 128, 256, 257, 384, 385, 512, 514, 640, 642, 1024, 1025, 1026, 1028, 1029, 1030

Offset

1,3

Comments

A finite topology is a finite set of finite sets closed under union and intersection and containing {} and the vertex set.

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 finite topologies by number of points is given by A000798.

Links

Table of n, a(n) for n=1..60.

Wikipedia Topological space

Example

The sequence of all finite topologies without their empty set together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  32: {{2,3}}
  34: {{2},{2,3}}
  40: {{3},{2,3}}
  42: {{2},{3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  69: {{1},{1,2},{1,2,3}}

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 100], SubsetQ[bpe/@bpe[#], Union[Union@@@Tuples[bpe/@bpe[#], 2], DeleteCases[Intersection@@@Tuples[bpe/@bpe[#], 2], {}]]]&]

Crossrefs

Cf. A000798, A001930, A003465, A048793, A102894, A102896, A326031, A326872, A326875, A326878.

Sequence in context: A326853 A326879 A326875 * A026486 A103838 A139283

Adjacent sequences: A326873 A326874 A326875 * A326877 A326878 A326879

Keyword

nonn

Author

Gus Wiseman, Jul 29 2019

Status

approved