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%I #4 Dec 10 2019 20:01:09
%S 0,5,12,180,35636,13
%N Least BII-number of a set-system with n distinct representatives obtainable by permuting the vertices.
%C A set-system is a finite set of finite nonempty sets of positive integers.
%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 set-systems together with their BII-numbers begins:
%e 0: {}
%e 5: {{1},{1,2}}
%e 12: {{1,2},{3}}
%e 180: {{1,2},{1,3},{2,3},{4}}
%e 35636: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{5}}
%e 13: {{1},{1,2},{3}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t graprms[m_]:=Union[Table[Sort[Sort/@(m/.Apply[Rule,Table[{p[[i]],i},{i,Length[p]}],{1}])],{p,Permutations[Union@@m]}]];
%t dv=Table[Length[graprms[bpe/@bpe[n]]],{n,0,1000}];
%t Table[Position[dv,i][[1,1]]-1,{i,First[Split[Union[dv],#1+1==#2&]]}]
%Y Positions of first appearances in A330231.
%Y The MM-number version is A330230.
%Y Achiral set-systems are counted by A083323.
%Y BII-numbers of fully chiral set-systems are A330226.
%Y Cf. A000120, A003238, A007716, A016031, A048793, A055621, A070939, A214577, A326031, A326702, A330098, A330101, A330195, A330217, A330229, A330233.
%K nonn
%O 1,2
%A _Gus Wiseman_, Dec 09 2019
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