A326873 BII-numbers of connectedness systems without singletons.

0, 4, 16, 32, 64, 68, 80, 84, 96, 100, 112, 116, 256, 288, 512, 528, 1024, 1028, 1280, 1284, 1536, 1540, 1792, 1796, 2048, 2052, 4096, 4112, 4352, 4368, 6144, 6160, 6400, 6416, 8192, 8224, 8704, 8736, 10240, 10272, 10752, 10784, 16384, 16388, 16400, 16416

Offset: 1

Author

Gus Wiseman, Jul 29 2019

Keywords

• nonn

Comments

We define a connectedness system (investigated by Vim van Dam in 2002) to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges.

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 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.

The enumeration of these set-systems by number of covered vertices is given by A326877.

Examples

			The sequence of all connectedness systems without singletons together with their BII-numbers begins:
     0: {}
     4: {{1,2}}
    16: {{1,3}}
    32: {{2,3}}
    64: {{1,2,3}}
    68: {{1,2},{1,2,3}}
    80: {{1,3},{1,2,3}}
    84: {{1,2},{1,3},{1,2,3}}
    96: {{2,3},{1,2,3}}
   100: {{1,2},{2,3},{1,2,3}}
   112: {{1,3},{2,3},{1,2,3}}
   116: {{1,2},{1,3},{2,3},{1,2,3}}
   256: {{1,4}}
   288: {{2,3},{1,4}}
   512: {{2,4}}
   528: {{1,3},{2,4}}
  1024: {{1,2,4}}
  1028: {{1,2},{1,2,4}}
  1280: {{1,4},{1,2,4}}
  1284: {{1,2},{1,4},{1,2,4}}
		

Links

• Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Crossrefs

Connectedness systems without singletons are counted by A072446, with unlabeled case A072444.

Connectedness systems are counted by A326866, with unlabeled case A326867.

BII-numbers of connectedness systems are A326872.

The connected case is A326879.

Cf. A029931, A048793, A072447, A326031, A326749, A326750, A326870, A326875, A326877.

Programs

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
connnosQ[eds_]:=!MemberQ[Length/@eds,1]&&SubsetQ[eds,Union@@@Select[Tuples[eds,2],Intersection@@#!={}&]];
Select[Range[0,1000],connnosQ[bpe/@bpe[#]]&]