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

Author

Gus Wiseman, Dec 10 2019

Keywords

• nonn

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.

Examples

			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}
		

Crossrefs

BII-numbers of set partitions are A326701.

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

Programs

Mathematica

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

Formula

Equal the complement of A000079 in A326701.