OFFSET

1,2

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.

LINKS

John Tyler Rascoe, Table of n, a(n) for n = 1..2000

Wikipedia, Axiom of choice.

EXAMPLE

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}}

MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

Select[Range[100], Length[Select[Tuples[bpe/@bpe[#]], UnsameQ@@#&]]==1&]

PROG

(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

CROSSREFS

These set-systems are counted by A367904.

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

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.

KEYWORD

nonn,base

AUTHOR

Gus Wiseman, Dec 11 2023

STATUS

approved