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

%I A326875 #10 Mar 06 2025 21:02:12
%S A326875 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,
%T A326875 80,81,82,84,85,86,87,88,89,92,93,96,97,98,100,101,102,103,104,106,
%U A326875 108,110,112,113,114,116,117,118,119,120,121,122,124,125,126,127,128
%N A326875 BII-numbers of set-systems that are closed under union.
%C A326875 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.
%C A326875 The enumeration of these set-systems by number of covered vertices is A102896.
%H A326875 John Tyler Rascoe, <a href="/A326875/b326875.txt">Table of n, a(n) for n = 1..3000</a>
%e A326875 The sequence of all set-systems that are closed under union together with their BII-numbers begins:
%e A326875    0: {}
%e A326875    1: {{1}}
%e A326875    2: {{2}}
%e A326875    4: {{1,2}}
%e A326875    5: {{1},{1,2}}
%e A326875    6: {{2},{1,2}}
%e A326875    7: {{1},{2},{1,2}}
%e A326875    8: {{3}}
%e A326875   16: {{1,3}}
%e A326875   17: {{1},{1,3}}
%e A326875   24: {{3},{1,3}}
%e A326875   25: {{1},{3},{1,3}}
%e A326875   32: {{2,3}}
%e A326875   34: {{2},{2,3}}
%e A326875   40: {{3},{2,3}}
%e A326875   42: {{2},{3},{2,3}}
%e A326875   64: {{1,2,3}}
%e A326875   65: {{1},{1,2,3}}
%e A326875   66: {{2},{1,2,3}}
%e A326875   68: {{1,2},{1,2,3}}
%e A326875   69: {{1},{1,2},{1,2,3}}
%e A326875   70: {{2},{1,2},{1,2,3}}
%e A326875   71: {{1},{2},{1,2},{1,2,3}}
%e A326875   72: {{3},{1,2,3}}
%e A326875   76: {{1,2},{3},{1,2,3}}
%e A326875   80: {{1,3},{1,2,3}}
%e A326875   81: {{1},{1,3},{1,2,3}}
%e A326875   82: {{2},{1,3},{1,2,3}}
%e A326875   84: {{1,2},{1,3},{1,2,3}}
%e A326875   85: {{1},{1,2},{1,3},{1,2,3}}
%e A326875   86: {{2},{1,2},{1,3},{1,2,3}}
%t A326875 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326875 Select[Range[0,100],SubsetQ[bpe/@bpe[#],Union@@@Tuples[bpe/@bpe[#],2]]&]
%o A326875 (Python)
%o A326875 from itertools import count, islice, combinations
%o A326875 def bin_i(n): #binary indices
%o A326875     return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
%o A326875 def a_gen():
%o A326875     for n in count(0):
%o A326875         E,f = [bin_i(k) for k in bin_i(n)],0
%o A326875         for i in combinations(E,2):
%o A326875             if list(set(i[0])|set(i[1])) not in E:
%o A326875                 f += 1
%o A326875                 break
%o A326875         if f < 1:
%o A326875             yield n
%o A326875 A326875_list = list(islice(a_gen(), 100)) # _John Tyler Rascoe_, Mar 06 2025
%Y A326875 Cf. A006126, A048793, A102894, A102896, A102897, A326031, A326872, A326874, A326876, A326880.
%K A326875 nonn,base
%O A326875 1,3
%A A326875 _Gus Wiseman_, Jul 29 2019