

A326701


BIInumbers of set partitions.


26



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
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OFFSET

1,3


COMMENTS

A binary index of n is any position of a 1 in its reversed binary expansion. We define the setsystem with BIInumber 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

Table of n, a(n) for n=1..54.


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[#]&]


CROSSREFS

MMnumbers of set partitions are A302521.
BIInumbers of chains of nonempty sets are A326703.
BIInumbers of antichains of nonempty sets are A326704.
Cf. A000120, A029931, A035327, A048793, A070939, A291166, A326031, A326675, A326702.
Sequence in context: A326704 A309314 A309326 * A061887 A005455 A047338
Adjacent sequences: A326698 A326699 A326700 * A326702 A326703 A326704


KEYWORD

nonn


AUTHOR

Gus Wiseman, Jul 21 2019


STATUS

approved



