A330195 - OEIS

%I #7 Dec 05 2019 17:42:21

%S 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 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 52,53,53,55,21,23,23,31,53,55,55,63,64,65,65,67,68,69

%N BII-number of the BII-normalization of the set-system with BII-number n.

%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.

%C 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 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   Brute-force:    2067: {{1},{2},{1,3},{3,4}}

%C   Lexicographic:   165: {{1},{4},{1,2},{2,3}}

%C   VDD:             525: {{1},{3},{1,2},{2,4}}

%C   MM:              270: {{2},{3},{1,2},{1,4}}

%C   BII:             150: {{2},{4},{1,2},{1,3}}

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Idempotent">Idempotence</a>

%F a(n) <= n.

%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t fbi[q_]:=If[q=={},0,Total[2^q]/2];

%t 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 brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];

%t Table[fbi[fbi/@biinorm[bpe/@bpe[n]]],{n,0,100}]

%Y This sequence is idempotent and its image/fixed points are A330109.

%Y A subset of A326754.

%Y Unlabeled spanning set-systems counted by vertices are A055621.

%Y Unlabeled set-systems counted by weight are A283877.

%Y BII-weight is A326031.

%Y Cf. A000120, A000612, A007716, A048793, A070939, A319559, A330061, A330101, A330102, A330194.

%Y Other fixed points:

%Y - Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).

%Y - Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).

%Y - VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).

%Y - MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).

%Y - BII: A330109 (set-systems).

%K nonn

%O 0,4

%A _Gus Wiseman_, Dec 05 2019