cp's OEIS Frontend

A330101 BII-number of the brute-force normalization of the set-system with BII-number n.

%I A330101 #11 Dec 13 2019 10:32:03
%S A330101 0,1,1,3,4,5,5,7,1,3,3,11,33,19,19,15,4,5,33,19,20,21,37,23,5,7,19,15,
%T A330101 37,23,51,31,4,33,5,19,20,37,21,23,5,19,7,15,37,51,23,31,20,37,37,51,
%U A330101 52,53,53,55,21,23,23,31,53,55,55,63,64,65,65,67,68,69,69
%N A330101 BII-number of the brute-force normalization of the set-system with BII-number n.
%C A330101 First differs from A330102 at a(148) = 545, A330102(148) = 274, with corresponding set-systems 545: {{1},{2,3},{2,4}} and 274: {{2},{1,3},{1,4}}.
%C A330101 A set-system is a finite set of finite nonempty sets of positive integers.
%C A330101 We define the brute-force 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 least representative, where the ordering of sets is first by length and then lexicographically.
%C A330101 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 A330101   Brute-force:    2067: {{1},{2},{1,3},{3,4}}
%C A330101   Lexicographic:   165: {{1},{4},{1,2},{2,3}}
%C A330101   VDD:             525: {{1},{3},{1,2},{2,4}}
%C A330101   MM:              270: {{2},{3},{1,2},{1,4}}
%C A330101   BII:             150: {{2},{4},{1,2},{1,3}}
%C A330101 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.
%H A330101 Wikipedia, <a href="https://en.wikipedia.org/wiki/Idempotent">Idempotence</a>
%t A330101 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A330101 fbi[q_]:=If[q=={},0,Total[2^q]/2];
%t A330101 brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];
%t A330101 brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
%t A330101 Table[fbi[fbi/@brute[bpe/@bpe[n]]],{n,0,100}]
%Y A330101 This sequence is idempotent and its image and fixed points are A330099.
%Y A330101 Non-isomorphic multiset partitions are A007716.
%Y A330101 Unlabeled spanning set-systems by vertices are A055621.
%Y A330101 Unlabeled set-systems by weight are A283877.
%Y A330101 Cf. A000612, A300913, A321405, A330061, A330102, A330105.
%Y A330101 Other fixed points:
%Y A330101 - Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
%Y A330101 - Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
%Y A330101 - VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
%Y A330101 - MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
%Y A330101 - BII: A330109 (set-systems).
%K A330101 nonn
%O A330101 0,4
%A A330101 _Gus Wiseman_, Dec 02 2019