A326879 BII-numbers of connected connectedness systems.

0, 1, 2, 4, 5, 6, 7, 8, 16, 17, 24, 25, 32, 34, 40, 42, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112

Offset

1,3

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. It is connected if it contains an edge containing all the vertices.

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 connected connectedness systems by number of vertices is given by A326868.

Links

Table of n, a(n) for n=1..65.

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

Example

The sequence of all connected connectedness systems together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  32: {{2,3}}
  34: {{2},{2,3}}
  40: {{3},{2,3}}
  42: {{2},{3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  67: {{1},{2},{1,2,3}}
  68: {{1,2},{1,2,3}}

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
connsysQ[eds_]:=SubsetQ[eds, Union@@@Select[Tuples[eds, 2], Intersection@@#!={}&]];
Select[Range[0, 100], #==0||MemberQ[bpe/@bpe[#], Union@@bpe/@bpe[#]]&&connsysQ[bpe/@bpe[#]]&]

Crossrefs

Connected connectedness systems are counted by A326868, with unlabeled version A326869.

Connected connectedness systems without singletons are counted by A072447.

The not necessarily connected case is A326872.

Cf. A029931, A048793, A072445, A072446, A326031, A326749, A326753, A326866, A326867, A326870, A326876.

Sequence in context: A326749 A327111 A326853 * A326875 A326876 A026486

Adjacent sequences: A326876 A326877 A326878 * A326880 A326881 A326882

Keyword

nonn

Author

Gus Wiseman, Jul 29 2019

Status

approved