cp's OEIS Frontend

A330102 BII-number of the VDD-normalization of the set-system with BII-number n.

%I A330102 #6 Dec 05 2019 08:23:47
%S A330102 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 A330102 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 A330102 52,53,53,55,21,23,23,31,53,55,55,63,64,65,65,67,68,69,69
%N A330102 BII-number of the VDD-normalization of the set-system with BII-number n.
%C A330102 First differs from A330101 at a(148) = 274, A330101(148) = 545, with corresponding set-systems 274: {{2},{1,3},{1,4}} and 545: {{1},{2,3},{2,4}}.
%C A330102 A set-system is a finite set of finite nonempty sets of positive integers.
%C A330102 We define the VDD (vertex-degrees decreasing) 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, then selecting only the representatives whose vertex-degrees are weakly decreasing, and finally taking the least of these representatives, where the ordering of sets is first by length and then lexicographically.
%C A330102 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 A330102   Brute-force:    2067: {{1},{2},{1,3},{3,4}}
%C A330102   Lexicographic:   165: {{1},{4},{1,2},{2,3}}
%C A330102   VDD:             525: {{1},{3},{1,2},{2,4}}
%C A330102   MM:              270: {{2},{3},{1,2},{1,4}}
%C A330102   BII:             150: {{2},{4},{1,2},{1,3}}
%C A330102 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.
%H A330102 Wikipedia, <a href="https://en.wikipedia.org/wiki/Idempotent">Idempotence</a>
%e A330102 56 is the BII-number of {{3},{1,3},{2,3}}, which has VDD-normalization {{1},{1,2},{1,3}} with BII-number 21, so a(56) = 21.
%t A330102 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A330102 fbi[q_]:=If[q=={},0,Total[2^q]/2];
%t A330102 sysnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];
%t A330102 sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
%t A330102 Table[fbi[fbi/@sysnorm[bpe/@bpe[n]]],{n,0,100}]
%Y A330102 This sequence is idempotent and its image/fixed points are A330100.
%Y A330102 Non-isomorphic multiset partitions are A007716.
%Y A330102 Unlabeled spanning set-systems counted by vertices are A055621.
%Y A330102 Unlabeled set-systems counted by weight are A283877.
%Y A330102 Cf. A000120, A000612, A048793, A070939, A300913, A319559, A321405, A326031, A326754, A330061, A330101.
%Y A330102 Other fixed points:
%Y A330102 - Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
%Y A330102 - Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
%Y A330102 - VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
%Y A330102 - MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
%Y A330102 - BII: A330109 (set-systems).
%K A330102 nonn
%O A330102 0,4
%A A330102 _Gus Wiseman_, Dec 04 2019