|
|
A367906
|
|
Numbers k such that it is possible to choose a different binary index of each binary index of k.
|
|
53
|
|
|
1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 26, 28, 32, 33, 34, 35, 36, 37, 38, 40, 41, 44, 48, 49, 50, 52, 56, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 76, 80, 81, 82, 84, 88, 96, 97, 98, 100, 104, 112, 128, 129, 130, 131, 132
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Also BII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice.
A binary index of k (row k 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 k to be obtained by taking the binary indices of each binary index of k. 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
|
|
|
EXAMPLE
|
The set-system {{2,3},{1,2,3},{1,4}} with BII-number 352 has choices such as (2,1,4) that satisfy the axiom, so 352 is in the sequence.
The terms together with the corresponding set-systems begin:
1: {{1}}
2: {{2}}
3: {{1},{2}}
4: {{1,2}}
5: {{1},{1,2}}
6: {{2},{1,2}}
8: {{3}}
9: {{1},{3}}
10: {{2},{3}}
11: {{1},{2},{3}}
12: {{1,2},{3}}
13: {{1},{1,2},{3}}
14: {{2},{1,2},{3}}
16: {{1,3}}
17: {{1},{1,3}}
|
|
MATHEMATICA
|
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[100], Select[Tuples[bpe/@bpe[#]], UnsameQ@@#&]!={}&]
|
|
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):
for j in list(product(*[bin_i(k) for k in bin_i(n)])):
if len(set(j)) == len(j):
yield(n); break
|
|
CROSSREFS
|
Unlabeled multiset partitions of this type are A368098, complement A368097.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
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).
|
|
KEYWORD
|
base,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|