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A326701 BII-numbers of set partitions.

%I A326701 #10 May 24 2024 16:18:59
%S A326701 0,1,2,3,4,8,9,10,11,12,16,18,32,33,64,128,129,130,131,132,136,137,
%T A326701 138,139,140,144,146,160,161,192,256,258,264,266,288,512,513,520,521,
%U A326701 528,1024,1032,2048,2049,2050,2051,2052,4096,4098,8192,8193,16384,32768,32769
%N A326701 BII-numbers of set partitions.
%C A326701 A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. 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, and {{2},{1,3}} is a set partition, it follows that 18 belongs to the sequence.
%H A326701 John Tyler Rascoe, <a href="/A326701/b326701.txt">Table of n, a(n) for n = 1..10000</a>
%e A326701 The sequence of all set partitions together with their BII numbers begins:
%e A326701     0: {}
%e A326701     1: {{1}}
%e A326701     2: {{2}}
%e A326701     3: {{1},{2}}
%e A326701     4: {{1,2}}
%e A326701     8: {{3}}
%e A326701     9: {{1},{3}}
%e A326701    10: {{2},{3}}
%e A326701    11: {{1},{2},{3}}
%e A326701    12: {{1,2},{3}}
%e A326701    16: {{1,3}}
%e A326701    18: {{2},{1,3}}
%e A326701    32: {{2,3}}
%e A326701    33: {{1},{2,3}}
%e A326701    64: {{1,2,3}}
%e A326701   128: {{4}}
%e A326701   129: {{1},{4}}
%e A326701   130: {{2},{4}}
%e A326701   131: {{1},{2},{4}}
%e A326701   132: {{1,2},{4}}
%e A326701   136: {{3},{4}}
%t A326701 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326701 Select[Range[0,1000],UnsameQ@@Join@@bpe/@bpe[#]&]
%o A326701 (Python)
%o A326701 from itertools import chain, count, combinations, islice
%o A326701 from sympy.utilities.iterables import multiset_partitions
%o A326701 def a_gen():
%o A326701     yield 0
%o A326701     for n in count(1):
%o A326701         t = []
%o A326701         for i in chain.from_iterable(combinations(range(1,n+1),r) for r in range(n+1)):
%o A326701             if n in i:
%o A326701                 for j in multiset_partitions(i):
%o A326701                     t.append(sum(2**(sum(2**(m-1) for m in k)-1) for k in j))
%o A326701         yield from sorted(t)
%o A326701 A326701_list = list(islice(a_gen(), 100)) # _John Tyler Rascoe_, May 24 2024
%Y A326701 MM-numbers of set partitions are A302521.
%Y A326701 BII-numbers of chains of nonempty sets are A326703.
%Y A326701 BII-numbers of antichains of nonempty sets are A326704.
%Y A326701 Cf. A000120, A029931, A035327, A048793, A070939, A291166, A326031, A326675, A326702.
%K A326701 nonn,base
%O A326701 1,3
%A A326701 _Gus Wiseman_, Jul 21 2019