A367908 Numbers n such that there is only one way to choose a different binary index of each binary index of n.

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, 56, 67, 69, 70, 73, 74, 81, 88, 98, 104, 128, 129, 130, 131, 133, 134, 136, 137, 138, 139, 141, 142, 145, 147, 149, 150, 152, 154, 156, 162, 163, 165, 166, 168

Offset: 1

Author

Gus Wiseman, Dec 11 2023

Keywords

• nonn
• base

Comments

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

Examples

			The set-system {{1},{1,2},{1,3}} with BII-number 21 satisfies the axiom in exactly one way, namely (1,2,3), so 21 is in the sequence.
The terms together with the corresponding set-systems begin:
   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}}
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 1..2000
• Wikipedia, Axiom of choice.

Crossrefs

These set-systems are counted by A367904.

Positions of 1's 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 are multiple choices we get A367909, counted by A367772.

The version for MM-numbers of multiset partitions is A368101.

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

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

A059201 counts covering T_0 set-systems.

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 for axiom, complement A368097.

Cf. A000612, A059519, A309326, A326675, A326702, A326753, A326872, A355529, A367902, A367912.

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).

Programs

Mathematica

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

Python

from itertools import count, islice, product
def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen(): #generator of terms
    for n in count(1):
        p = list(product(*[bin_i(k) for k in bin_i(n)]))
        x,c = len(p),0
        for j in range(x):
            if len(set(p[j])) == len(p[j]): c += 1
            if j+1 == x and c == 1: yield(n)
A367908_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Feb 10 2024

Formula

A367907 U A367908 U A367909 = A000027.