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A330195 BII-number of the BII-normalization of the set-system with BII-number n.

%I A330195 #7 Dec 05 2019 17:42:21
%S A330195 0,1,1,3,4,5,5,7,1,3,3,11,12,13,13,15,4,5,12,13,20,21,22,23,5,7,13,15,
%T A330195 22,23,30,31,4,12,5,13,20,22,21,23,5,13,7,15,22,30,23,31,20,22,22,30,
%U A330195 52,53,53,55,21,23,23,31,53,55,55,63,64,65,65,67,68,69
%N A330195 BII-number of the BII-normalization of the set-system with BII-number n.
%C A330195 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.
%C A330195 We define the BII-normalization of a set-system to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the representative with the smallest BII-number.
%C A330195 For example, 156 is the BII-number of {{3},{4},{1,2},{1,3}}, which has the following normalizations, together with their BII-numbers:
%C A330195   Brute-force:    2067: {{1},{2},{1,3},{3,4}}
%C A330195   Lexicographic:   165: {{1},{4},{1,2},{2,3}}
%C A330195   VDD:             525: {{1},{3},{1,2},{2,4}}
%C A330195   MM:              270: {{2},{3},{1,2},{1,4}}
%C A330195   BII:             150: {{2},{4},{1,2},{1,3}}
%H A330195 Wikipedia, <a href="https://en.wikipedia.org/wiki/Idempotent">Idempotence</a>
%F A330195 a(n) <= n.
%t A330195 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A330195 fbi[q_]:=If[q=={},0,Total[2^q]/2];
%t A330195 biinorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],biinorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[SortBy[brute[m,1],fbi[fbi/@#]&]]];
%t A330195 brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
%t A330195 Table[fbi[fbi/@biinorm[bpe/@bpe[n]]],{n,0,100}]
%Y A330195 This sequence is idempotent and its image/fixed points are A330109.
%Y A330195 A subset of A326754.
%Y A330195 Unlabeled spanning set-systems counted by vertices are A055621.
%Y A330195 Unlabeled set-systems counted by weight are A283877.
%Y A330195 BII-weight is A326031.
%Y A330195 Cf. A000120, A000612, A007716, A048793, A070939, A319559, A330061, A330101, A330102, A330194.
%Y A330195 Other fixed points:
%Y A330195 - Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
%Y A330195 - Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
%Y A330195 - VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
%Y A330195 - MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
%Y A330195 - BII: A330109 (set-systems).
%K A330195 nonn
%O A330195 0,4
%A A330195 _Gus Wiseman_, Dec 05 2019