%I #4 Dec 10 2019 20:01:35
%S 3,9,10,11,12,18,33,129,130,131,132,136,137,138,139,140,144,146,160,
%T 161,192,258,264,266,288,513,520,521,528,1032,2049,2050,2051,2052,
%U 4098,8193,32769,32770,32771,32772,32776,32777,32778,32779,32780,32784,32786,32800
%N BII-numbers of set partitions with at least two blocks.
%C 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.
%F Equal the complement of A000079 in A326701.
%e The sequence of all set partitions with at least two parts together with their BII-numbers begins:
%e 3: {1}{2} 140: {3}{4}{12} 2049: {1}{34}
%e 9: {1}{3} 144: {4}{13} 2050: {2}{34}
%e 10: {2}{3} 146: {2}{4}{13} 2051: {1}{2}{34}
%e 11: {1}{2}{3} 160: {4}{23} 2052: {12}{34}
%e 12: {3}{12} 161: {1}{4}{23} 4098: {2}{134}
%e 18: {2}{13} 192: {4}{123} 8193: {1}{234}
%e 33: {1}{23} 258: {2}{14} 32769: {1}{5}
%e 129: {1}{4} 264: {3}{14} 32770: {2}{5}
%e 130: {2}{4} 266: {2}{3}{14} 32771: {1}{2}{5}
%e 131: {1}{2}{4} 288: {14}{23} 32772: {5}{12}
%e 132: {4}{12} 513: {1}{24} 32776: {3}{5}
%e 136: {3}{4} 520: {3}{24} 32777: {1}{3}{5}
%e 137: {1}{3}{4} 521: {1}{3}{24} 32778: {2}{3}{5}
%e 138: {2}{3}{4} 528: {13}{24} 32779: {1}{2}{3}{5}
%e 139: {1}{2}{3}{4} 1032: {3}{124} 32780: {3}{5}{12}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t Select[Range[1000],Length[bpe[#]]>=2&&Length[Join@@bpe/@bpe[#]]==Length[Union@@bpe/@bpe[#]]&]
%Y BII-numbers of set partitions are A326701.
%Y Cf. A048793, A070939, A072639, A326031, A326753, A329661.
%K nonn
%O 1,1
%A _Gus Wiseman_, Dec 10 2019