A367905 -id:A367905 - cp's OEIS frontend

A367911 Sorted positions of first appearances in A367905.

1, 4, 7, 20, 68, 320, 352, 1088, 3136, 5184, 13376, 16704, 17472, 70720, 82240, 83008, 90112, 90176

Offset: 1

Gus Wiseman, Dec 16 2023

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

			The terms together with the corresponding set-systems begin:
      1: {{1}}
      4: {{1,2}}
      7: {{1},{2},{1,2}}
     20: {{1,2},{1,3}}
     68: {{1,2},{1,2,3}}
    320: {{1,2,3},{1,4}}
    352: {{2,3},{1,2,3},{1,4}}
   1088: {{1,2,3},{1,2,4}}
   3136: {{1,2,3},{1,2,4},{3,4}}
   5184: {{1,2,3},{1,2,4},{1,3,4}}
  13376: {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  16704: {{1,2,3},{1,4},{1,2,3,4}}
  17472: {{1,2,3},{1,2,4},{1,2,3,4}}
  70720: {{1,2,3},{1,2,4},{1,3,4},{1,5}}
  82240: {{1,2,3},{1,4},{1,2,3,4},{1,5}}

Sorted positions of first appearances in A367905.
The unsorted version is A367910.
Multisets without distinctness are A367915, unsorted A367913.
Without distinctness we have A368112, unsorted A368111.
For sets instead of sequences we have A368185, unsorted A368184.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326753, A367902, A367906, A367907, A367912, A368109, A368183.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
c=Table[Length[Select[Tuples[bpe/@bpe[n]],UnsameQ@@#&]],{n,1000}];
Select[Range[Length[c]],FreeQ[Take[c,#-1],c[[#]]]&]

A367903 Number of sets of nonempty subsets of {1..n} contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 1, 67, 30997, 2147296425, 9223372036784737528, 170141183460469231731687303625772608225, 57896044618658097711785492504343953926634992332820282019728791606173188627779

Offset: 0

Gus Wiseman, Dec 05 2023

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.

			The a(2) = 1 set-system is {{1},{2},{1,2}}.
The a(3) = 67 set-systems have following 21 non-isomorphic representatives:
  {{1},{2},{1,2}}
  {{1},{2},{3},{1,2}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,2},{1,3}}
  {{1},{2},{1,2},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{1},{2},{1,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3}}
  {{1},{1,2},{1,3},{1,2,3}}
  {{1},{1,2},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3}}
  {{1},{2},{3},{1,2},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3}}
  {{1},{2},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,3},{2,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Wikipedia, Axiom of choice.

Multisets of multisets of this type are ranked by A355529.
The version without singletons is A367769.
The version for simple graphs is A367867, covering A367868.
The version allowing empty edges is A367901.
The complement is A367902, without singletons A367770, ranks A367906.
For a unique choice (instead of none) we have A367904, ranks A367908.
These set-systems have ranks A367907.
An unlabeled version is A368094, for multiset partitions A368097.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems.
A326031 gives weight of the set-system with BII-number n.
Cf. A007716, A083323, A092918, A102896, A283877, A306445, A355739, A355740, A367862, A367905, A368409, A368413.

Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]], Select[Tuples[#],UnsameQ@@#&]=={}&]],{n,0,3}]

a(n) + A367904(n) + A367772(n) = A058891(n+1) = 2^(2^n-1).

a(5)-a(8) from Christian Sievers, Jul 26 2024

A367902 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 2, 7, 61, 1771, 187223, 70038280, 90111497503, 397783376192189

Offset: 0

Gus Wiseman, Dec 05 2023

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.

			The a(2) = 7 set-systems:
  {}
  {{1}}
  {{2}}
  {{1,2}}
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}

Wikipedia, Axiom of choice.

The version for simple graphs is A133686, covering A367869.
The version without singletons is A367770.
The complement allowing empty edges is A367901.
The complement is A367903, without singletons A367769, ranks A367907.
For a unique choice we have A367904, ranks A367908.
These set-systems have ranks A367906.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems.
A326031 gives weight of the set-system with BII-number n.
Cf. A007716, A083323, A092918, A102896, A283877, A306445, A355739, A355740, A367862, A367905, A370636.

Table[Length[Select[Subsets[Subsets[Range[n]]], Select[Tuples[#],UnsameQ@@#&]!={}&]],{n,0,3}]

a(n) = A370636(2^n-1). - Alois P. Heinz, Mar 09 2024

a(6)-a(8) from Christian Sievers, Jul 25 2024

A368100 Numbers of which it is possible to choose a different prime factor of each prime index.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 119, 123, 127, 129, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163

Offset: 1

Gus Wiseman, Dec 12 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 2849 are {4,5,12}, with prime factors {{2,2},{5},{2,2,3}}, and of the two choices (2,5,2) and (2,5,3) the latter has all different terms, so 2849 is in the sequence.
The terms together with their prime indices of prime indices begin:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  29: {{1,3}}
  31: {{5}}
  33: {{1},{3}}
  35: {{2},{1,1}}
  37: {{1,1,2}}
  39: {{1},{1,2}}

The complement is A355529, odd A355535, binary A367907.
Positions of positive terms in A367771.
The version for binary indices is A367906, positive positions in A367905.
For a unique choice we have A368101, binary A367908.
The version for divisors instead of factors is A368110, complement A355740.
A058891 counts set-systems, covering A003465, connected A323818.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
Cf. A092918,  A355737, A355739, A355741, A355744, A355745, A367902.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], Select[Tuples[prix/@prix[#]], UnsameQ@@#&]!={}&]

A367907 Numbers n such that it is not possible to choose a different binary index of each binary index of n.

Original entry on oeis.org

7, 15, 23, 25, 27, 29, 30, 31, 39, 42, 43, 45, 46, 47, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 71, 75, 77, 78, 79, 83, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 99, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115, 116, 117, 118, 119, 120, 121

Offset: 1

Gus Wiseman, Dec 11 2023

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

			The set-system {{1},{2},{1,2},{1,3}} with BII-number 23 has choices (1,2,1,1), (1,2,1,3), (1,2,2,1), (1,2,2,3), but none of these has all different elements, so 23 is in the sequence.
The terms together with the corresponding set-systems begin:
   7: {{1},{2},{1,2}}
  15: {{1},{2},{1,2},{3}}
  23: {{1},{2},{1,2},{1,3}}
  25: {{1},{3},{1,3}}
  27: {{1},{2},{3},{1,3}}
  29: {{1},{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  31: {{1},{2},{1,2},{3},{1,3}}
  39: {{1},{2},{1,2},{2,3}}
  42: {{2},{3},{2,3}}
  43: {{1},{2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  46: {{2},{1,2},{3},{2,3}}
  47: {{1},{2},{1,2},{3},{2,3}}
  51: {{1},{2},{1,3},{2,3}}

John Tyler Rascoe, Table of n, a(n) for n = 1..10000
Wikipedia, Axiom of choice.

These set-systems are counted by A367903, non-isomorphic A368094.
Positions of zeros in A367905, firsts A367910, sorted A367911.
The complement is A367906.
If there is one unique choice we get A367908, counted by A367904.
If there are multiple choices we get A367909, counted by A367772.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
Cf. A000612, A055621, A072639, A083323, A309326, A326702, A326753, A367769, A367901, 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).

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

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 = len(p)
        for j in range(x):
            if len(set(p[j])) == len(p[j]): break
            if j+1 == x: yield(n)
A367907_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Feb 10 2024

A367907 U A367908 U A367909 = A000027.

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 11 2023

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.

			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}}

John Tyler Rascoe, Table of n, a(n) for n = 1..10000
Wikipedia, Axiom of choice.

These set-systems are counted by A367902, non-isomorphic A368095.
Positions of positive terms in A367905, firsts A367910, sorted A367911.
The complement is A367907.
If there is one unique choice we get A367908, counted by A367904.
If there are multiple choices we get A367909, counted by A367772.
Unlabeled multiset partitions of this type are A368098, complement A368097.
A version for MM-numbers of multisets is A368100, complement A355529.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
Cf. A000612, A055621, A059519, A072639, A083323, A309326, A326702, A326753, A367770, 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).

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

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
A367906_list = list(islice(a_gen(),100)) # John Tyler Rascoe, Dec 23 2023

A368097 Number of non-isomorphic multiset partitions of weight n contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 1, 3, 12, 37, 133, 433, 1516, 5209, 18555

Offset: 0

Gus Wiseman, Dec 25 2023

A multiset partition is a finite multiset of finite nonempty multisets. The weight of a multiset partition is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

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.

			Non-isomorphic representatives of the a(2) = 1 through a(4) = 12 multiset partitions:
  {{1},{1}}  {{1},{1,1}}    {{1},{1,1,1}}
             {{1},{1},{1}}  {{1,1},{1,1}}
             {{1},{2},{2}}  {{1},{1},{1,1}}
                            {{1},{1},{2,2}}
                            {{1},{1},{2,3}}
                            {{1},{2},{1,2}}
                            {{1},{2},{2,2}}
                            {{2},{2},{1,2}}
                            {{1},{1},{1},{1}}
                            {{1},{1},{2},{2}}
                            {{1},{2},{2},{2}}
                            {{1},{2},{3},{3}}

Wikipedia, Axiom of choice.

The case of unlabeled graphs appears to be A140637, complement A134964.
These multiset partitions have ranks A355529.
The case of labeled graphs is A367867, complement A133686.
Set-systems not of this type are A367902, ranks A367906.
Set-systems of this type are A367903, ranks A367907.
For set-systems we have A368094, complement A368095.
The complement is A368098, ranks A368100, connected case A368412.
Minimal multiset partitions of this type are ranked by A368187.
The connected case is A368411.
Factorizations of this type are counted by A368413, complement A368414.
For set multipartitions we have A368421, complement A368422.
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A302545, A316983, A317533, A319616, A321405, A367905, A368409, A368410.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort/@(#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]], {s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute/@Select[mpm[n], Select[Tuples[#],UnsameQ@@#&]=={}&]]], {n,0,6}]

A370592 Number of integer partitions of n such that it is possible to choose a different prime factor of each part.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 3, 3, 4, 4, 5, 6, 7, 9, 11, 12, 12, 16, 18, 22, 26, 29, 29, 37, 41, 49, 55, 61, 68, 72, 88, 98, 110, 120, 135, 146, 166, 190, 209, 227, 252, 277, 309, 346, 379, 413, 447, 500, 548, 606, 665, 727, 785, 857, 949, 1033, 1132, 1228, 1328, 1440

Offset: 0

Gus Wiseman, Feb 29 2024

nonn

			The partition (10,6,4) has choice (5,3,2) so is counted under a(20).
The a(0) = 1 through a(10) = 4 partitions:
  ()  .  (2)  (3)  (4)  (5)    (6)  (7)    (8)    (9)    (10)
                        (3,2)       (4,3)  (5,3)  (5,4)  (6,4)
                                    (5,2)  (6,2)  (6,3)  (7,3)
                                                  (7,2)  (5,3,2)
The a(0) = 1 through a(17) = 12 partitions (0 = {}, A..H = 10..17):
  0  .  2  3  4  5   6  7   8   9   A    B   C    D    E    F    G    H
                 32     43  53  54  64   65  66   76   86   87   97   98
                        52  62  63  73   74  75   85   95   96   A6   A7
                                72  532  83  A2   94   A4   A5   B5   B6
                                         92  543  A3   B3   B4   C4   C5
                                             732  B2   C2   C3   D3   D4
                                                  652  653  D2   E2   E3
                                                       743  654  754  F2
                                                       752  753  763  665
                                                            762  853  764
                                                            A32  952  A43
                                                                 B32  7532

The version for divisors instead of factors is A239312, ranks A368110.
The version for set-systems is A367902, ranks A367906, unlabeled A368095.
The complement for set-systems is A367903, ranks A367907, unlabeled A368094.
For unlabeled multiset partitions we have A368098, complement A368097.
These partitions have ranks A368100.
The version for factorizations is A368414, complement A368413.
The complement is counted by A370593, ranks A355529.
For a unique choice we have A370594, ranks A370647.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A133686, A355739, A355740, A355745, A367771, A367905, A370585, A370586, A370636.

Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]>0&]],{n,0,30}]

a(n) = A000041(n) - A370593(n).

A368094 Number of non-isomorphic set-systems of weight n contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 5, 12, 36, 97, 291

Offset: 0

Gus Wiseman, Dec 23 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

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.

			Non-isomorphic representatives of the a(5) = 1 through a(7) = 12 set-systems:
  {{1},{2},{3},{2,3}}  {{1},{2},{1,3},{2,3}}    {{1},{2},{1,2},{3,4,5}}
                       {{1},{2},{3},{1,2,3}}    {{1},{3},{2,3},{1,2,3}}
                       {{2},{3},{1,3},{2,3}}    {{1},{4},{1,4},{2,3,4}}
                       {{3},{4},{1,2},{3,4}}    {{2},{3},{2,3},{1,2,3}}
                       {{1},{2},{3},{4},{3,4}}  {{3},{1,2},{1,3},{2,3}}
                                                {{1},{2},{3},{1,3},{2,3}}
                                                {{1},{2},{3},{2,4},{3,4}}
                                                {{1},{2},{3},{4},{2,3,4}}
                                                {{1},{3},{4},{2,4},{3,4}}
                                                {{1},{4},{5},{2,3},{4,5}}
                                                {{2},{3},{4},{1,2},{3,4}}
                                                {{1},{2},{3},{4},{5},{4,5}}

Wikipedia, Axiom of choice.

The case of unlabeled graphs is A140637, complement A134964.
The case of labeled graphs is A367867, complement A133686.
The labeled version is A367903, ranks A367907.
The complement is counted by A368095, connected A368410.
Repeats allowed: A368097, ranks A355529, complement A368098, ranks A368100.
Minimal multiset partitions of this type are ranked by A368187.
The connected case is A368409.
Factorizations of this type are counted by A368413, complement A368414.
Allowing repeated edges gives A368421, complement A368422.
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A302545, A306005, A316983, A317533, A319616, A321155, A321405, A326031, A330223, A330227, A367905.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort/@(#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]], {s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute/@Select[mpm[n], UnsameQ@@#&&And@@UnsameQ@@@# && Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,8}]

A368109 Number of ways to choose a binary index of each binary index of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 4, 4, 4, 4, 8, 8, 8, 8, 3, 3, 3, 3, 6, 6, 6, 6, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6, 6, 6, 12, 12, 12

Offset: 0

Gus Wiseman, Dec 12 2023

nonn

First differs from A367912 at a(52) = 8, A367912(52) = 7.
A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.
Run-lengths are all 4 or 8.

			The binary indices of binary indices of 20 are {{1,2},{1,3}}, with choices (1,1), (1,3), (2,1), (2,3), so a(20) = 4.
The binary indices of binary indices of 52 are {{1,2},{1,3},{2,3}}, with choices (1,1,1), (1,1,3), (1,3,2), (1,3,3), (2,1,2), (2,1,3), (2,3,2), (2,3,3), so a(52) = 8.

All entries appear to belong to A003586.
Positions of ones are A253317.
The version for prime indices is A355741, for multisets A355744.
Choosing a multiset (not sequence) gives A367912, firsts A367913.
Positions of first appearances are A368111, sorted A368112.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326753, A355731, A355739, A367771, A367905, A367906, A367915.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]],1];
Table[Length[Tuples[bpe/@bpe[n]]], {n,0,100}]

a(n) = Product_{k in A048793(n)} A000120(k).

cp's OEIS Frontend

A367911 Sorted positions of first appearances in A367905.

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A367903 Number of sets of nonempty subsets of {1..n} contradicting a strict version of the axiom of choice.

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A367902 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice.

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A368100 Numbers of which it is possible to choose a different prime factor of each prime index.

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A367907 Numbers n such that it is not possible to choose a different binary index of each binary index of n.

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

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A368097 Number of non-isomorphic multiset partitions of weight n contradicting a strict version of the axiom of choice.

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A370592 Number of integer partitions of n such that it is possible to choose a different prime factor of each part.

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