A326701 BII-numbers of set partitions.

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

Author

Gus Wiseman, Jul 21 2019

Keywords

• nonn
• base

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.

Examples

			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}}
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 1..10000

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.

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

Programs

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,1000],UnsameQ@@Join@@bpe/@bpe[#]&]

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