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

%I A330296 #4 Dec 10 2019 20:01:35
%S A330296 3,9,10,11,12,18,33,129,130,131,132,136,137,138,139,140,144,146,160,
%T A330296 161,192,258,264,266,288,513,520,521,528,1032,2049,2050,2051,2052,
%U A330296 4098,8193,32769,32770,32771,32772,32776,32777,32778,32779,32780,32784,32786,32800
%N A330296 BII-numbers of set partitions with at least two blocks.
%C A330296 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 A330296 Equal the complement of A000079 in A326701.
%e A330296 The sequence of all set partitions with at least two parts together with their BII-numbers begins:
%e A330296     3: {1}{2}          140: {3}{4}{12}     2049: {1}{34}
%e A330296     9: {1}{3}          144: {4}{13}        2050: {2}{34}
%e A330296    10: {2}{3}          146: {2}{4}{13}     2051: {1}{2}{34}
%e A330296    11: {1}{2}{3}       160: {4}{23}        2052: {12}{34}
%e A330296    12: {3}{12}         161: {1}{4}{23}     4098: {2}{134}
%e A330296    18: {2}{13}         192: {4}{123}       8193: {1}{234}
%e A330296    33: {1}{23}         258: {2}{14}       32769: {1}{5}
%e A330296   129: {1}{4}          264: {3}{14}       32770: {2}{5}
%e A330296   130: {2}{4}          266: {2}{3}{14}    32771: {1}{2}{5}
%e A330296   131: {1}{2}{4}       288: {14}{23}      32772: {5}{12}
%e A330296   132: {4}{12}         513: {1}{24}       32776: {3}{5}
%e A330296   136: {3}{4}          520: {3}{24}       32777: {1}{3}{5}
%e A330296   137: {1}{3}{4}       521: {1}{3}{24}    32778: {2}{3}{5}
%e A330296   138: {2}{3}{4}       528: {13}{24}      32779: {1}{2}{3}{5}
%e A330296   139: {1}{2}{3}{4}   1032: {3}{124}      32780: {3}{5}{12}
%t A330296 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A330296 Select[Range[1000],Length[bpe[#]]>=2&&Length[Join@@bpe/@bpe[#]]==Length[Union@@bpe/@bpe[#]]&]
%Y A330296 BII-numbers of set partitions are A326701.
%Y A330296 Cf. A048793, A070939, A072639, A326031, A326753, A329661.
%K A330296 nonn
%O A330296 1,1
%A A330296 _Gus Wiseman_, Dec 10 2019