A327081 BII-numbers of maximal uniform set-systems covering an initial interval of positive integers.

1, 3, 4, 11, 52, 64, 139, 2868, 13376, 16384, 32907

Offset: 1

Author

Gus Wiseman, Aug 20 2019

Keywords

• nonn
• more

Comments

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 (finite set of finite nonempty sets) 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 set-system is uniform if all edges have the same size.

Examples

			The sequence of all maximal uniform set-systems covering an initial interval together with their BII-numbers begins:
      0: {}
      1: {{1}}
      3: {{1},{2}}
      4: {{1,2}}
     11: {{1},{2},{3}}
     52: {{1,2},{1,3},{2,3}}
     64: {{1,2,3}}
    139: {{1},{2},{3},{4}}
   2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
  13376: {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  16384: {{1,2,3,4}}
  32907: {{1},{2},{3},{4},{5}}
		

Crossrefs

BII-numbers of uniform set-systems are A326783.

BII-numbers of maximal uniform set-systems are A327080.

Cf. A000120, A048793, A070939, A326031, A326784, A326785, A327041.

Programs

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[1000],With[{sys=bpe/@bpe[#]},#==0||normQ[Union@@sys]&&SameQ@@Length/@sys&&Length[sys]==Binomial[Length[Union@@sys],Length[First[sys]]]]&]