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

%I A326880 #11 Mar 07 2025 21:30:04
%S A326880 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,
%T A326880 29,31,32,33,34,35,38,39,40,41,42,43,46,47,56,57,58,59,63,64,65,66,67,
%U A326880 68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,85,87,88
%N A326880 BII-numbers of set-systems that are closed under nonempty intersection.
%C A326880 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.
%C A326880 The enumeration of these set-systems by number of covered vertices is A326881.
%H A326880 John Tyler Rascoe, <a href="/A326880/b326880.txt">Table of n, a(n) for n = 1..10000</a>
%e A326880 Most small numbers are in the sequence, but the sequence of non-terms together with the set-systems with those BII-numbers begins:
%e A326880   20: {{1,2},{1,3}}
%e A326880   22: {{2},{1,2},{1,3}}
%e A326880   28: {{1,2},{3},{1,3}}
%e A326880   30: {{2},{1,2},{3},{1,3}}
%e A326880   36: {{1,2},{2,3}}
%e A326880   37: {{1},{1,2},{2,3}}
%e A326880   44: {{1,2},{3},{2,3}}
%e A326880   45: {{1},{1,2},{3},{2,3}}
%e A326880   48: {{1,3},{2,3}}
%e A326880   49: {{1},{1,3},{2,3}}
%e A326880   50: {{2},{1,3},{2,3}}
%e A326880   51: {{1},{2},{1,3},{2,3}}
%e A326880   52: {{1,2},{1,3},{2,3}}
%e A326880   53: {{1},{1,2},{1,3},{2,3}}
%e A326880   54: {{2},{1,2},{1,3},{2,3}}
%e A326880   55: {{1},{2},{1,2},{1,3},{2,3}}
%e A326880   60: {{1,2},{3},{1,3},{2,3}}
%e A326880   61: {{1},{1,2},{3},{1,3},{2,3}}
%e A326880   62: {{2},{1,2},{3},{1,3},{2,3}}
%e A326880   84: {{1,2},{1,3},{1,2,3}}
%t A326880 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326880 Select[Range[0,100],SubsetQ[bpe/@bpe[#],Intersection@@@Select[Tuples[bpe/@bpe[#],2],Intersection@@#!={}&]]&]
%o A326880 (Python)
%o A326880 from itertools import count, islice, combinations
%o A326880 def bin_i(n): #binary indices
%o A326880     return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
%o A326880 def a_gen():
%o A326880     for n in count(0):
%o A326880         E,f = [bin_i(k) for k in bin_i(n)],0
%o A326880         for i in combinations(E,2):
%o A326880             x = list(set(i[0])&set(i[1]))
%o A326880             if x not in E and len(x) > 0:
%o A326880                 f += 1
%o A326880                 break
%o A326880         if f < 1:
%o A326880             yield n
%o A326880 A326880_list = list(islice(a_gen(), 100)) # _John Tyler Rascoe_, Mar 07 2025
%Y A326880 Cf. A006126, A048793, A102894, A102895, A102896, A102897, A306445, A326031, A326872, A326874, A326875, A326876, A326881.
%K A326880 nonn,base
%O A326880 1,3
%A A326880 _Gus Wiseman_, Jul 29 2019