A330226 BII-numbers of fully chiral set-systems.

0, 1, 2, 5, 6, 8, 13, 14, 17, 19, 22, 23, 24, 26, 28, 29, 34, 35, 37, 39, 40, 41, 44, 46, 49, 50, 57, 58, 69, 70, 77, 78, 81, 83, 86, 87, 88, 90, 92, 93, 98, 99, 101, 103, 104, 105, 108, 110, 113, 114, 121, 122, 128, 133, 134, 145, 150, 151, 152, 156, 157, 162

Offset: 1

Author

Gus Wiseman, Dec 08 2019

Keywords

• nonn

Comments

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the vertices gives a different representative.

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.

Examples

			The sequence of all fully chiral set-systems together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  17: {{1},{1,3}}
  19: {{1},{2},{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}
  24: {{3},{1,3}}
  26: {{2},{3},{1,3}}
  28: {{3},{1,2},{1,3}}
  29: {{1},{3},{1,2},{1,3}}
  34: {{2},{2,3}}
  35: {{1},{2},{2,3}}
  37: {{1},{1,2},{2,3}}
  39: {{1},{2},{1,2},{2,3}}
For example, 28 is in the sequence because all six permutations give different representatives, namely:
  {{1},{1,2},{2,3}}
  {{1},{1,3},{2,3}}
  {{2},{1,2},{1,3}}
  {{2},{1,3},{2,3}}
  {{3},{1,2},{1,3}}
  {{3},{1,2},{2,3}}
		

Crossrefs

A subset of A326947.

Achiral set-systems are counted by A083323.

BII-numbers of achiral set-systems are A330217.

Non-isomorphic, fully chiral multiset partitions are A330227.

Fully chiral partitions are counted by A330228.

Fully chiral covering set-systems are A330229.

Fully chiral factorizations are A330235.

MM-numbers of fully chiral multisets of multisets are A330236.

Cf. A000120, A000612, A016031, A048793, A070939, A326031, A326702, A330218, A330231, A330232.

Programs

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Select[Range[0,100],Length[graprms[bpe/@bpe[#]]]==Length[Union@@bpe/@bpe[#]]!&]