A367917 BII-numbers of set-systems with the same number of edges as covered vertices.

0, 1, 2, 3, 5, 6, 8, 9, 10, 11, 13, 14, 17, 19, 21, 22, 24, 26, 28, 34, 35, 37, 38, 40, 41, 44, 49, 50, 52, 56, 67, 69, 70, 73, 74, 76, 81, 82, 84, 88, 97, 98, 100, 104, 112, 128, 129, 130, 131, 133, 134, 136, 137, 138, 139, 141, 142, 145, 147, 149, 150, 152

Offset

1,3

Comments

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. A set-system is a finite set of finite nonempty sets. 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.

Links

Table of n, a(n) for n=1..62.

Example

The terms together with the corresponding set-systems begin:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  17: {{1},{1,3}}
  19: {{1},{2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  24: {{3},{1,3}}
  26: {{2},{3},{1,3}}
  28: {{1,2},{3},{1,3}}
  34: {{2},{2,3}}
  35: {{1},{2},{2,3}}
  37: {{1},{1,2},{2,3}}

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 100], Length[bpe[#]]==Length[Union@@bpe/@bpe[#]]&]

Crossrefs

These set-systems are counted by A054780 and A367916, A368186.

Graphs of this type are A367862, covering A367863, unlabeled A006649.

A003465 counts set-systems covering {1..n}, unlabeled A055621.

A048793 lists binary indices, length A000120, sum A029931.

A058891 counts set-systems, connected A323818, unlabeled A000612.

A070939 gives length of binary expansion.

A136556 counts set-systems on {1..n} with n edges.

Cf. A057500, A059201, A072639, A096111, A116508, A309326, A326031, A326702, A326753, A326754, A367770, A367902, A367905.

Sequence in context: A047450 A039073 A026359 * A367908 A032953 A270813

Adjacent sequences: A367914 A367915 A367916 * A367918 A367919 A367920

Keyword

nonn

Author

Gus Wiseman, Dec 12 2023

Status

approved