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%I #8 Mar 26 2020 20:42:10
%S 0,1,2,5,6,8,13,14,17,19,22,23,24,26,28,29,34,35,37,39,40,41,44,46,49,
%T 50,57,58,69,70,77,78,81,83,86,87,88,90,92,93,98,99,101,103,104,105,
%U 108,110,113,114,121,122,128,133,134,145,150,151,152,156,157,162
%N BII-numbers of fully chiral set-systems.
%C A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the vertices gives a different representative.
%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 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 The sequence of all fully chiral set-systems together with their BII-numbers begins:
%e 0: {}
%e 1: {{1}}
%e 2: {{2}}
%e 5: {{1},{1,2}}
%e 6: {{2},{1,2}}
%e 8: {{3}}
%e 13: {{1},{1,2},{3}}
%e 14: {{2},{1,2},{3}}
%e 17: {{1},{1,3}}
%e 19: {{1},{2},{1,3}}
%e 22: {{2},{1,2},{1,3}}
%e 23: {{1},{2},{1,2},{1,3}}
%e 24: {{3},{1,3}}
%e 26: {{2},{3},{1,3}}
%e 28: {{3},{1,2},{1,3}}
%e 29: {{1},{3},{1,2},{1,3}}
%e 34: {{2},{2,3}}
%e 35: {{1},{2},{2,3}}
%e 37: {{1},{1,2},{2,3}}
%e 39: {{1},{2},{1,2},{2,3}}
%e For example, 28 is in the sequence because all six permutations give different representatives, namely:
%e {{1},{1,2},{2,3}}
%e {{1},{1,3},{2,3}}
%e {{2},{1,2},{1,3}}
%e {{2},{1,3},{2,3}}
%e {{3},{1,2},{1,3}}
%e {{3},{1,2},{2,3}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
%t Select[Range[0,100],Length[graprms[bpe/@bpe[#]]]==Length[Union@@bpe/@bpe[#]]!&]
%Y A subset of A326947.
%Y Achiral set-systems are counted by A083323.
%Y BII-numbers of achiral set-systems are A330217.
%Y Non-isomorphic, fully chiral multiset partitions are A330227.
%Y Fully chiral partitions are counted by A330228.
%Y Fully chiral covering set-systems are A330229.
%Y Fully chiral factorizations are A330235.
%Y MM-numbers of fully chiral multisets of multisets are A330236.
%Y Cf. A000120, A000612, A016031, A048793, A070939, A326031, A326702, A330218, A330231, A330232.
%K nonn
%O 1,3
%A _Gus Wiseman_, Dec 08 2019
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