A367909 Numbers n such that there is more than one way to choose a different binary index of each binary index of n.

4, 12, 16, 18, 20, 32, 33, 36, 48, 52, 64, 65, 66, 68, 72, 76, 80, 82, 84, 96, 97, 100, 112, 132, 140, 144, 146, 148, 160, 161, 164, 176, 180, 192, 193, 194, 196, 200, 204, 208, 210, 212, 224, 225, 228, 240, 256, 258, 260, 264, 266, 268, 272, 274, 276, 288

Offset

1,1

Comments

Also BII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice in more than one way.

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 digits (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.

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

Links

Table of n, a(n) for n=1..56.

Wikipedia, Axiom of choice.

Formula

A367907 U A367908 U A367909 = A000027.

Example

The set-system {{1},{1,2},{1,3}} with BII-number 21 satisfies the axiom in only one way (1,2,3), so 21 is not in the sequence.
The terms together with the corresponding set-systems begin:
   4: {{1,2}}
  12: {{1,2},{3}}
  16: {{1,3}}
  18: {{2},{1,3}}
  20: {{1,2},{1,3}}
  32: {{2,3}}
  33: {{1},{2,3}}
  36: {{1,2},{2,3}}
  48: {{1,3},{2,3}}
  52: {{1,2},{1,3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  72: {{3},{1,2,3}}

Mathematica

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

Crossrefs

These set-systems are counted by A367772.

Positions of terms > 1 in A367905, firsts A367910, sorted firsts A367911.

If there is at least one choice we get A367906, counted by A367902.

If there are no choices we get A367907, counted by A367903.

If there is one unique choice we get A367908, counted by A367904.

A048793 lists binary indices, length A000120, reverse A272020, sum A029931.

A058891 counts set-systems, covering A003465, connected A323818.

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

A326031 gives weight of the set-system with BII-number n.

A368098 counts unlabeled multiset partitions per axiom, complement A368097.

Cf. A000612, A055621, A072639, A309326, A326702, A326753, A355529, A368100.

BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers), A326783 (uniform), A326784 (regular), A326788 (simple), A330217 (achiral).

Sequence in context: A368715 A144976 A119622 * A280892 A328840 A255411

Adjacent sequences: A367906 A367907 A367908 * A367910 A367911 A367912

Keyword

nonn

Author

Gus Wiseman, Dec 11 2023

Status

approved