A330296 BII-numbers of set partitions with at least two blocks.

3, 9, 10, 11, 12, 18, 33, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 160, 161, 192, 258, 264, 266, 288, 513, 520, 521, 528, 1032, 2049, 2050, 2051, 2052, 4098, 8193, 32769, 32770, 32771, 32772, 32776, 32777, 32778, 32779, 32780, 32784, 32786, 32800

Offset

1,1

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets of positive integers) has a different BII-number. 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, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

Links

Table of n, a(n) for n=1..48.

Formula

Equal the complement of A000079 in A326701.

Example

The sequence of all set partitions with at least two parts together with their BII-numbers begins:
    3: {1}{2}          140: {3}{4}{12}     2049: {1}{34}
    9: {1}{3}          144: {4}{13}        2050: {2}{34}
   10: {2}{3}          146: {2}{4}{13}     2051: {1}{2}{34}
   11: {1}{2}{3}       160: {4}{23}        2052: {12}{34}
   12: {3}{12}         161: {1}{4}{23}     4098: {2}{134}
   18: {2}{13}         192: {4}{123}       8193: {1}{234}
   33: {1}{23}         258: {2}{14}       32769: {1}{5}
  129: {1}{4}          264: {3}{14}       32770: {2}{5}
  130: {2}{4}          266: {2}{3}{14}    32771: {1}{2}{5}
  131: {1}{2}{4}       288: {14}{23}      32772: {5}{12}
  132: {4}{12}         513: {1}{24}       32776: {3}{5}
  136: {3}{4}          520: {3}{24}       32777: {1}{3}{5}
  137: {1}{3}{4}       521: {1}{3}{24}    32778: {2}{3}{5}
  138: {2}{3}{4}       528: {13}{24}      32779: {1}{2}{3}{5}
  139: {1}{2}{3}{4}   1032: {3}{124}      32780: {3}{5}{12}

Mathematica

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

Crossrefs

BII-numbers of set partitions are A326701.

Cf. A048793, A070939, A072639, A326031, A326753, A329661.

Sequence in context: A114022 A181862 A329560 * A043048 A308421 A309349

Adjacent sequences: A330293 A330294 A330295 * A330297 A330298 A330299

Keyword

nonn

Author

Gus Wiseman, Dec 10 2019

Status

approved