A326875 BII-numbers of set-systems that are closed under union.

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, 84, 85, 86, 87, 88, 89, 92, 93, 96, 97, 98, 100, 101, 102, 103, 104, 106, 108, 110, 112, 113, 114, 116, 117, 118, 119, 120, 121, 122, 124, 125, 126, 127, 128

Offset: 1

Author

Gus Wiseman, Jul 29 2019

Keywords

• nonn
• base

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.

The enumeration of these set-systems by number of covered vertices is A102896.

Examples

			The sequence of all set-systems that are closed under union 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}}
  70: {{2},{1,2},{1,2,3}}
  71: {{1},{2},{1,2},{1,2,3}}
  72: {{3},{1,2,3}}
  76: {{1,2},{3},{1,2,3}}
  80: {{1,3},{1,2,3}}
  81: {{1},{1,3},{1,2,3}}
  82: {{2},{1,3},{1,2,3}}
  84: {{1,2},{1,3},{1,2,3}}
  85: {{1},{1,2},{1,3},{1,2,3}}
  86: {{2},{1,2},{1,3},{1,2,3}}
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 1..3000

Crossrefs

Cf. A006126, A048793, A102894, A102896, A102897, A326031, A326872, A326874, A326876, A326880.

Programs

Mathematica

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

Python

from itertools import count, islice, combinations
def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen():
    for n in count(0):
        E,f = [bin_i(k) for k in bin_i(n)],0
        for i in combinations(E,2):
            if list(set(i[0])|set(i[1])) not in E:
                f += 1
                break
        if f < 1:
            yield n
A326875_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Mar 06 2025