cp's OEIS Frontend

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

%I A367908 #16 Feb 11 2024 02:34:53
%S A367908 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,
%T A367908 49,50,56,67,69,70,73,74,81,88,98,104,128,129,130,131,133,134,136,137,
%U A367908 138,139,141,142,145,147,149,150,152,154,156,162,163,165,166,168
%N A367908 Numbers n such that there is only one way to choose a different binary index of each binary index of n.
%C A367908 Also BII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice in exactly one way.
%C A367908 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.
%C A367908 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.
%H A367908 John Tyler Rascoe, <a href="/A367908/b367908.txt">Table of n, a(n) for n = 1..2000</a>
%H A367908 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%F A367908 A367907 U A367908 U A367909 = A000027.
%e A367908 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.
%e A367908 The terms together with the corresponding set-systems begin:
%e A367908    1: {{1}}
%e A367908    2: {{2}}
%e A367908    3: {{1},{2}}
%e A367908    5: {{1},{1,2}}
%e A367908    6: {{2},{1,2}}
%e A367908    8: {{3}}
%e A367908    9: {{1},{3}}
%e A367908   10: {{2},{3}}
%e A367908   11: {{1},{2},{3}}
%e A367908   13: {{1},{1,2},{3}}
%e A367908   14: {{2},{1,2},{3}}
%e A367908   17: {{1},{1,3}}
%e A367908   19: {{1},{2},{1,3}}
%e A367908   21: {{1},{1,2},{1,3}}
%e A367908   22: {{2},{1,2},{1,3}}
%t A367908 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A367908 Select[Range[100], Length[Select[Tuples[bpe/@bpe[#]], UnsameQ@@#&]]==1&]
%o A367908 (Python)
%o A367908 from itertools import count, islice, product
%o A367908 def bin_i(n): #binary indices
%o A367908     return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
%o A367908 def a_gen(): #generator of terms
%o A367908     for n in count(1):
%o A367908         p = list(product(*[bin_i(k) for k in bin_i(n)]))
%o A367908         x,c = len(p),0
%o A367908         for j in range(x):
%o A367908             if len(set(p[j])) == len(p[j]): c += 1
%o A367908             if j+1 == x and c == 1: yield(n)
%o A367908 A367908_list = list(islice(a_gen(), 100)) # _John Tyler Rascoe_, Feb 10 2024
%Y A367908 These set-systems are counted by A367904.
%Y A367908 Positions of 1's in A367905, firsts A367910, sorted firsts A367911.
%Y A367908 If there is at least one choice we get A367906, counted by A367902.
%Y A367908 If there are no choices we get A367907, counted by A367903.
%Y A367908 If there are multiple choices we get A367909, counted by A367772.
%Y A367908 The version for MM-numbers of multiset partitions is A368101.
%Y A367908 A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
%Y A367908 A058891 counts set-systems, covering A003465, connected A323818.
%Y A367908 A059201 counts covering T_0 set-systems.
%Y A367908 A070939 gives length of binary expansion.
%Y A367908 A096111 gives product of binary indices.
%Y A367908 A326031 gives weight of the set-system with BII-number n.
%Y A367908 A368098 counts unlabeled multiset partitions for axiom, complement A368097.
%Y A367908 Cf. A000612, A059519, A309326, A326675, A326702, A326753, A326872, A355529, A367902, A367912.
%Y A367908 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).
%K A367908 nonn,base
%O A367908 1,2
%A A367908 _Gus Wiseman_, Dec 11 2023