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A326701
BII-numbers of set partitions.
33
0, 1, 2, 3, 4, 8, 9, 10, 11, 12, 16, 18, 32, 33, 64, 128, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 160, 161, 192, 256, 258, 264, 266, 288, 512, 513, 520, 521, 528, 1024, 1032, 2048, 2049, 2050, 2051, 2052, 4096, 4098, 8192, 8193, 16384, 32768, 32769
OFFSET
1,3
COMMENTS
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.
LINKS
EXAMPLE
The sequence of all set partitions together with their BII numbers begins:
0: {}
1: {{1}}
2: {{2}}
3: {{1},{2}}
4: {{1,2}}
8: {{3}}
9: {{1},{3}}
10: {{2},{3}}
11: {{1},{2},{3}}
12: {{1,2},{3}}
16: {{1,3}}
18: {{2},{1,3}}
32: {{2,3}}
33: {{1},{2,3}}
64: {{1,2,3}}
128: {{4}}
129: {{1},{4}}
130: {{2},{4}}
131: {{1},{2},{4}}
132: {{1,2},{4}}
136: {{3},{4}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 1000], UnsameQ@@Join@@bpe/@bpe[#]&]
PROG
(Python)
from itertools import chain, count, combinations, islice
from sympy.utilities.iterables import multiset_partitions
def a_gen():
yield 0
for n in count(1):
t = []
for i in chain.from_iterable(combinations(range(1, n+1), r) for r in range(n+1)):
if n in i:
for j in multiset_partitions(i):
t.append(sum(2**(sum(2**(m-1) for m in k)-1) for k in j))
yield from sorted(t)
A326701_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, May 24 2024
CROSSREFS
MM-numbers of set partitions are A302521.
BII-numbers of chains of nonempty sets are A326703.
BII-numbers of antichains of nonempty sets are A326704.
Sequence in context: A326704 A309314 A309326 * A061887 A005455 A047338
KEYWORD
nonn,base
AUTHOR
Gus Wiseman, Jul 21 2019
STATUS
approved