A330099 BII-numbers of brute-force normalized set-systems.

0, 1, 3, 4, 5, 7, 11, 15, 19, 20, 21, 23, 31, 33, 37, 51, 52, 53, 55, 63, 64, 65, 67, 68, 69, 71, 75, 79, 83, 84, 85, 87, 95, 97, 101, 115, 116, 117, 119, 127, 139, 143, 159, 191, 203, 207, 223, 255, 267, 271, 275, 276, 277, 279, 287, 307, 308, 309, 311, 319, 331

Offset: 1

Author

Gus Wiseman, Dec 02 2019

Keywords

• nonn
• eigen

Comments

First differs from A330100 in having 545 and lacking 179, with corresponding set-systems 545: {{1},{2,3},{2,4}} and 179: {{1},{2},{4},{1,3},{2,3}}.

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

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.

For example, 156 is the BII-number of {{3},{4},{1,2},{1,3}}, which has the following normalizations, together with their BII-numbers:

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

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

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

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

BII: 150: {{2},{4},{1,2},{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.

There are A055621(n) entries m such that A326702(m) = n, where A326702(k) is the number of covered vertices in the set-system with BII-number k.

There are A283877(n) entries m such that A326031(m) = n, where A326031(k) is the weight of the set-system with BII-number k.

Examples

			The sequence of all nonempty brute-force normalized set-systems together with their BII-numbers begins:
   1: {1}                  52: {12}{13}{23}
   3: {1}{2}               53: {1}{12}{13}{23}
   4: {12}                 55: {1}{2}{12}{13}{23}
   5: {1}{12}              63: {1}{2}{3}{12}{13}{23}
   7: {1}{2}{12}           64: {123}
  11: {1}{2}{3}            65: {1}{123}
  15: {1}{2}{3}{12}        67: {1}{2}{123}
  19: {1}{2}{13}           68: {12}{123}
  20: {12}{13}             69: {1}{12}{123}
  21: {1}{12}{13}          71: {1}{2}{12}{123}
  23: {1}{2}{12}{13}       75: {1}{2}{3}{123}
  31: {1}{2}{3}{12}{13}    79: {1}{2}{3}{12}{123}
  33: {1}{23}              83: {1}{2}{13}{123}
  37: {1}{12}{23}          84: {12}{13}{123}
  51: {1}{2}{13}{23}       85: {1}{12}{13}{123}
		

Crossrefs

Equals the image/fixed points of the idempotent sequence A330101.

Non-isomorphic multiset partitions are A007716.

Unlabeled spanning set-systems by span are A055621.

Unlabeled spanning set-systems by weight are A283877.

Cf. A000612, A300913, A321405, A330061, A330102, A330105.

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];
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]]]];
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[#]]==brute[bpe/@bpe[#]]&]