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%I A330217 #8 Dec 08 2019 20:54:40
%S A330217 0,1,2,3,4,7,8,9,10,11,16,25,32,42,52,63,64,75,116,127,128,129,130,
%T A330217 131,136,137,138,139,256,385,512,642,772,903,1024,1155,1796,1927,2048,
%U A330217 2184,2320,2457,2592,2730,2868,3007,4096,4233,6416,6553,8192,8330
%N A330217 BII-numbers of achiral set-systems.
%C A330217 A set-system is a finite set of finite nonempty sets. It is achiral if it is not changed by any permutation of the vertices.
%C A330217 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 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.
%e A330217 The sequence of all achiral set-systems together with their BII-numbers begins:
%e A330217    1: {{1}}
%e A330217    2: {{2}}
%e A330217    3: {{1},{2}}
%e A330217    4: {{1,2}}
%e A330217    7: {{1},{2},{1,2}}
%e A330217    8: {{3}}
%e A330217    9: {{1},{3}}
%e A330217   10: {{2},{3}}
%e A330217   11: {{1},{2},{3}}
%e A330217   16: {{1,3}}
%e A330217   25: {{1},{3},{1,3}}
%e A330217   32: {{2,3}}
%e A330217   42: {{2},{3},{2,3}}
%e A330217   52: {{1,2},{1,3},{2,3}}
%e A330217   63: {{1},{2},{3},{1,2},{1,3},{2,3}}
%e A330217   64: {{1,2,3}}
%e A330217   75: {{1},{2},{3},{1,2,3}}
%t A330217 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A330217 graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
%t A330217 Select[Range[0,1000],Length[graprms[bpe/@bpe[#]]]==1&]
%Y A330217 These are numbers n such that A330231(n) = 1.
%Y A330217 Achiral set-systems are counted by A083323.
%Y A330217 MG-numbers of planted achiral trees are A214577.
%Y A330217 Non-isomorphic achiral multiset partitions are A330223.
%Y A330217 Achiral integer partitions are counted by A330224.
%Y A330217 BII-numbers of fully chiral set-systems are A330226.
%Y A330217 MM-numbers of achiral multisets of multisets are A330232.
%Y A330217 Achiral factorizations are A330234.
%Y A330217 Cf. A000120, A003238, A016031, A048793, A070939, A326031, A326702, A327080, A327081, A330218, A330229, A330233.
%K A330217 nonn
%O A330217 1,3
%A A330217 _Gus Wiseman_, Dec 06 2019

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