A326701 - OEIS

%I #10 May 24 2024 16:18:59

%S 0,1,2,3,4,8,9,10,11,12,16,18,32,33,64,128,129,130,131,132,136,137,

%T 138,139,140,144,146,160,161,192,256,258,264,266,288,512,513,520,521,

%U 528,1024,1032,2048,2049,2050,2051,2052,4096,4098,8192,8193,16384,32768,32769

%N BII-numbers of set partitions.

%C 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 John Tyler Rascoe, <a href="/A326701/b326701.txt">Table of n, a(n) for n = 1..10000</a>

%e The sequence of all set partitions together with their BII numbers begins:

%e     0: {}

%e     1: {{1}}

%e     2: {{2}}

%e     3: {{1},{2}}

%e     4: {{1,2}}

%e     8: {{3}}

%e     9: {{1},{3}}

%e    10: {{2},{3}}

%e    11: {{1},{2},{3}}

%e    12: {{1,2},{3}}

%e    16: {{1,3}}

%e    18: {{2},{1,3}}

%e    32: {{2,3}}

%e    33: {{1},{2,3}}

%e    64: {{1,2,3}}

%e   128: {{4}}

%e   129: {{1},{4}}

%e   130: {{2},{4}}

%e   131: {{1},{2},{4}}

%e   132: {{1,2},{4}}

%e   136: {{3},{4}}

%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t Select[Range[0,1000],UnsameQ@@Join@@bpe/@bpe[#]&]

%o (Python)

%o from itertools import chain, count, combinations, islice

%o from sympy.utilities.iterables import multiset_partitions

%o def a_gen():

%o     yield 0

%o     for n in count(1):

%o         t = []

%o         for i in chain.from_iterable(combinations(range(1,n+1),r) for r in range(n+1)):

%o             if n in i:

%o                 for j in multiset_partitions(i):

%o                     t.append(sum(2**(sum(2**(m-1) for m in k)-1) for k in j))

%o         yield from sorted(t)

%o A326701_list = list(islice(a_gen(), 100)) # _John Tyler Rascoe_, May 24 2024

%Y MM-numbers of set partitions are A302521.

%Y BII-numbers of chains of nonempty sets are A326703.

%Y BII-numbers of antichains of nonempty sets are A326704.

%Y Cf. A000120, A029931, A035327, A048793, A070939, A291166, A326031, A326675, A326702.

%K nonn,base

%O 1,3

%A _Gus Wiseman_, Jul 21 2019