cp's OEIS Frontend

A326754 BII-numbers of set-systems covering an initial interval of positive integers.

%I A326754 #10 Jun 21 2024 02:06:15
%S A326754 0,1,3,4,5,6,7,11,12,13,14,15,18,19,20,21,22,23,26,27,28,29,30,31,33,
%T A326754 35,36,37,38,39,41,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,
%U A326754 60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78
%N A326754 BII-numbers of set-systems covering an initial interval of positive integers.
%C A326754 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.
%H A326754 John Tyler Rascoe, <a href="/A326754/b326754.txt">Table of n, a(n) for n = 1..10000</a>
%e A326754 The sequence of all covering set-systems together with their BII-numbers begins:
%e A326754    0: {}
%e A326754    1: {{1}}
%e A326754    3: {{1},{2}}
%e A326754    4: {{1,2}}
%e A326754    5: {{1},{1,2}}
%e A326754    6: {{2},{1,2}}
%e A326754    7: {{1},{2},{1,2}}
%e A326754   11: {{1},{2},{3}}
%e A326754   12: {{1,2},{3}}
%e A326754   13: {{1},{1,2},{3}}
%e A326754   14: {{2},{1,2},{3}}
%e A326754   15: {{1},{2},{1,2},{3}}
%e A326754   18: {{2},{1,3}}
%e A326754   19: {{1},{2},{1,3}}
%e A326754   20: {{1,2},{1,3}}
%e A326754   21: {{1},{1,2},{1,3}}
%e A326754   22: {{2},{1,2},{1,3}}
%e A326754   23: {{1},{2},{1,2},{1,3}}
%e A326754   26: {{2},{3},{1,3}}
%e A326754   27: {{1},{2},{3},{1,3}}
%e A326754   28: {{1,2},{3},{1,3}}
%e A326754   29: {{1},{1,2},{3},{1,3}}
%e A326754   30: {{2},{1,2},{3},{1,3}}
%t A326754 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326754 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A326754 Select[Range[0,100],normQ[Join@@bpe/@bpe[#]]&]
%o A326754 (Python)
%o A326754 from itertools import chain, count, islice
%o A326754 def bin_i(n): #binary indices
%o A326754     return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
%o A326754 def a_gen():
%o A326754     for n in count(0):
%o A326754         s = set(i for i in chain.from_iterable([bin_i(k) for k in bin_i(n)]))
%o A326754         y = len(s)
%o A326754         if sum(s) == (y*(y+1))//2:
%o A326754             yield n
%o A326754 A326754_list = list(islice(a_gen(), 100)) # _John Tyler Rascoe_, Jun 20 2024
%Y A326754 Cf. A000120, A003465, A006126, A048793, A070939, A293510, A326031, A326702.
%Y A326754 Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees).
%K A326754 nonn,base
%O A326754 1,3
%A A326754 _Gus Wiseman_, Jul 23 2019