A326905 BII-numbers of set-systems (without {}) closed under intersection.

0, 1, 2, 4, 5, 6, 8, 16, 17, 21, 24, 32, 34, 38, 40, 56, 64, 65, 66, 68, 69, 70, 72, 80, 81, 85, 88, 96, 98, 102, 104, 120, 128, 256, 257, 261, 273, 277, 321, 325, 337, 341, 384, 512, 514, 518, 546, 550, 578, 582, 610, 614, 640, 896, 1024, 1025, 1026, 1028

Offset

1,3

Comments

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.

Links

Table of n, a(n) for n=1..58.

Example

The sequence of all set-systems closed under intersection together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  21: {{1},{1,2},{1,3}}
  24: {{3},{1,3}}
  32: {{2,3}}
  34: {{2},{2,3}}
  38: {{2},{1,2},{2,3}}
  40: {{3},{2,3}}
  56: {{3},{1,3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}

Mathematica

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

Crossrefs

The case with union instead of intersection is A326875.

The case closed under union and intersection is A326913.

Set-systems closed under intersection and containing the vertex set are A326903.

Set-systems closed under intersection are A326901, with unlabeled version A326904.

Cf. A006058, A102895, A102898, A326866, A326876, A326878, A326882, A326900, A326902.

Sequence in context: A176654 A185867 A326910 * A327061 A326913 A326703

Adjacent sequences: A326902 A326903 A326904 * A326906 A326907 A326908

Keyword

nonn

Author

Gus Wiseman, Aug 04 2019

Status

approved