cp's OEIS Frontend

A367906 Numbers k such that it is possible to choose a different binary index of each binary index of k.

%I A367906 #14 Dec 28 2023 19:27:39
%S A367906 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,
%T A367906 34,35,36,37,38,40,41,44,48,49,50,52,56,64,65,66,67,68,69,70,72,73,74,
%U A367906 76,80,81,82,84,88,96,97,98,100,104,112,128,129,130,131,132
%N A367906 Numbers k such that it is possible to choose a different binary index of each binary index of k.
%C A367906 Also BII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice.
%C A367906 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.
%C A367906 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 A367906 John Tyler Rascoe, <a href="/A367906/b367906.txt">Table of n, a(n) for n = 1..10000</a>
%H A367906 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%e A367906 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.
%e A367906 The terms together with the corresponding set-systems begin:
%e A367906    1: {{1}}
%e A367906    2: {{2}}
%e A367906    3: {{1},{2}}
%e A367906    4: {{1,2}}
%e A367906    5: {{1},{1,2}}
%e A367906    6: {{2},{1,2}}
%e A367906    8: {{3}}
%e A367906    9: {{1},{3}}
%e A367906   10: {{2},{3}}
%e A367906   11: {{1},{2},{3}}
%e A367906   12: {{1,2},{3}}
%e A367906   13: {{1},{1,2},{3}}
%e A367906   14: {{2},{1,2},{3}}
%e A367906   16: {{1,3}}
%e A367906   17: {{1},{1,3}}
%t A367906 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A367906 Select[Range[100], Select[Tuples[bpe/@bpe[#]], UnsameQ@@#&]!={}&]
%o A367906 (Python)
%o A367906 from itertools import count, islice, product
%o A367906 def bin_i(n): #binary indices
%o A367906     return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
%o A367906 def a_gen(): #generator of terms
%o A367906     for n in count(1):
%o A367906         for j in list(product(*[bin_i(k) for k in bin_i(n)])):
%o A367906             if len(set(j)) == len(j):
%o A367906                 yield(n); break
%o A367906 A367906_list = list(islice(a_gen(),100)) # _John Tyler Rascoe_, Dec 23 2023
%Y A367906 These set-systems are counted by A367902, non-isomorphic A368095.
%Y A367906 Positions of positive terms in A367905, firsts A367910, sorted A367911.
%Y A367906 The complement is A367907.
%Y A367906 If there is one unique choice we get A367908, counted by A367904.
%Y A367906 If there are multiple choices we get A367909, counted by A367772.
%Y A367906 Unlabeled multiset partitions of this type are A368098, complement A368097.
%Y A367906 A version for MM-numbers of multisets is A368100, complement A355529.
%Y A367906 A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
%Y A367906 A058891 counts set-systems, A003465 covering, A323818 connected.
%Y A367906 A070939 gives length of binary expansion.
%Y A367906 A096111 gives product of binary indices.
%Y A367906 A326031 gives weight of the set-system with BII-number n.
%Y A367906 Cf. A000612, A055621, A059519, A072639, A083323, A309326, A326702, A326753, A367770, A367912.
%Y A367906 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 A367906 base,nonn
%O A367906 1,2
%A A367906 _Gus Wiseman_, Dec 11 2023