A330123 BII-numbers of MM-normalized set-systems.

0, 1, 3, 4, 5, 7, 11, 13, 15, 20, 21, 23, 31, 33, 37, 45, 52, 53, 55, 63, 64, 65, 67, 68, 69, 71, 75, 77, 79, 84, 85, 87, 95, 97, 101, 109, 116, 117, 119, 127, 139, 143, 159, 173, 180, 181, 183, 191, 195, 196, 197, 199, 203, 205, 207, 212, 213, 215, 223, 225, 229

Offset: 1

Author

Gus Wiseman, Dec 05 2019

Keywords

• nonn

Comments

First differs from A330110 in lacking 141 and having 180, with corresponding set-systems 141: {{1},{3},{4},{1,2}} and 180: {{4},{1,2},{1,3},{2,3}}.

A set-system is a finite set of finite nonempty set of positive integers.

We define the MM-normalization of a multiset of multisets 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 MM-number.

For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:

Brute-force: 43287: {{1},{2,3},{2,2,4}}

Lexicographic: 43143: {{1},{2,4},{2,2,3}}

VDD: 15515: {{2},{1,3},{1,1,4}}

MM: 15265: {{2},{1,4},{1,1,3}}

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.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

Examples

			The sequence of all MM-normalized set-systems together with their BII-numbers begins:
   0: {}                           45: {{1},{3},{1,2},{2,3}}
   1: {{1}}                        52: {{1,2},{1,3},{2,3}}
   3: {{1},{2}}                    53: {{1},{1,2},{1,3},{2,3}}
   4: {{1,2}}                      55: {{1},{2},{1,2},{1,3},{2,3}}
   5: {{1},{1,2}}                  63: {{1},{2},{3},{1,2},{1,3},{2,3}}
   7: {{1},{2},{1,2}}              64: {{1,2,3}}
  11: {{1},{2},{3}}                65: {{1},{1,2,3}}
  13: {{1},{3},{1,2}}              67: {{1},{2},{1,2,3}}
  15: {{1},{2},{3},{1,2}}          68: {{1,2},{1,2,3}}
  20: {{1,2},{1,3}}                69: {{1},{1,2},{1,2,3}}
  21: {{1},{1,2},{1,3}}            71: {{1},{2},{1,2},{1,2,3}}
  23: {{1},{2},{1,2},{1,3}}        75: {{1},{2},{3},{1,2,3}}
  31: {{1},{2},{3},{1,2},{1,3}}    77: {{1},{3},{1,2},{1,2,3}}
  33: {{1},{2,3}}                  79: {{1},{2},{3},{1,2},{1,2,3}}
  37: {{1},{1,2},{2,3}}            84: {{1,2},{1,3},{1,2,3}}
		

Crossrefs

A subset of A326754.

Non-isomorphic multiset partitions are A007716.

Unlabeled spanning set-systems counted by vertices are A055621.

Unlabeled set-systems counted by weight are A283877.

MM-weight is A302242.

Cf. A000120, A000612, A048793, A056239, A070939, A112798, A300913, A319559, A320456, A326031, A330101, A330102, A330194.

Other fixed points:

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

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

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

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

- BII: A330109 (set-systems).

Programs

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
mmnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],mmnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[SortBy[brute[m,1],Map[Times@@Prime/@#&,#,{0,1}]&]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Select[Range[0,100],Sort[bpe/@bpe[#]]==mmnorm[bpe/@bpe[#]]&]