A326880 BII-numbers of set-systems that are closed under nonempty intersection.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 38, 39, 40, 41, 42, 43, 46, 47, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 87, 88

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.

The enumeration of these set-systems by number of covered vertices is A326881.

Links

Table of n, a(n) for n=1..68.

Example

Most small numbers are in the sequence, but the sequence of non-terms together with the set-systems with those BII-numbers begins:
  20: {{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  28: {{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  36: {{1,2},{2,3}}
  37: {{1},{1,2},{2,3}}
  44: {{1,2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  48: {{1,3},{2,3}}
  49: {{1},{1,3},{2,3}}
  50: {{2},{1,3},{2,3}}
  51: {{1},{2},{1,3},{2,3}}
  52: {{1,2},{1,3},{2,3}}
  53: {{1},{1,2},{1,3},{2,3}}
  54: {{2},{1,2},{1,3},{2,3}}
  55: {{1},{2},{1,2},{1,3},{2,3}}
  60: {{1,2},{3},{1,3},{2,3}}
  61: {{1},{1,2},{3},{1,3},{2,3}}
  62: {{2},{1,2},{3},{1,3},{2,3}}
  84: {{1,2},{1,3},{1,2,3}}

Mathematica

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

Crossrefs

Cf. A006126, A048793, A102894, A102895, A102896, A102897, A306445, A326031, A326872, A326874, A326875, A326876, A326881.

Sequence in context: A320249 A309066 A084981 * A359077 A078453 A052425

Adjacent sequences: A326877 A326878 A326879 * A326881 A326882 A326883

Keyword

nonn

Author

Gus Wiseman, Jul 29 2019

Status

approved