A367907 -id:A367907 - cp's OEIS frontend

A368600 Number of ways to choose a set of n nonempty subsets of {1..n} such that it is not possible to choose a different element from each.

0, 0, 0, 3, 164, 18625, 5491851, 4649088885, 12219849683346

Offset: 0

Gus Wiseman, Jan 01 2024

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(3) = 3 set-systems:
  {{1},{2},{1,2}}
  {{1},{3},{1,3}}
  {{2},{3},{2,3}}

Wikipedia, Axiom of choice.

For a unique choice we have A003024, any length A367904 (ranks A367908).
Sets of n nonempty subsets of {1..n} are counted by A136556.
For any length we have A367903, ranks A367907, no singletons A367769.
The complement is A368601, any length A367902 (see also A367770, 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, unlabeled A323819.
Cf. A003025, A088957, A133686, A334282, A355529, A355740, A367862, A367867, A367868, A367901, A368094, A368097.

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

from itertools import combinations, product, chain
from scipy.special import comb
def v(c):
    for elements in product(*c):
        if len(set(elements)) == len(elements):
            return True
    return False
def a(n):
    if n == 0:
        return 1
    subsets = list(chain.from_iterable(combinations(range(1, n + 1), r) for r in range(1, n + 1)))
    cs = combinations(subsets, n)
    c = sum(1 for c in cs if v(c))
    return c
[print(int(comb(2**n-1,n) - a(n))) for n in range(7)] # Robert P. P. McKone, Jan 02 2024

a(n) = A136556(n) - A368601(n).

a(6) from Robert P. P. McKone, Jan 02 2024

a(7)-a(8) from Christian Sievers, Jul 25 2024

A368601 Number of ways to choose a set of n nonempty subsets of {1..n} such that it is possible to choose a different element from each.

Original entry on oeis.org

1, 1, 3, 32, 1201, 151286, 62453670, 84707326890, 384641855115279

Offset: 0

Gus Wiseman, Jan 01 2024

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) = 3 set-systems:
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
Non-isomorphic representatives of the a(3) = 32 set-systems:
  {{1},{2},{3}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{1,3}}
  {{1},{1,2},{2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}

Wikipedia, Axiom of choice.

For a unique choice we have A003024, any length A367904 (ranks A367908).
Sets of n nonempty subsets of {1..n} are counted by A136556.
For any length we have A367902, ranks A367906, no singletons A367770.
The complement is A368600, any length A367903 (see also A367907, A367769).
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, unlabeled A323819.
Cf. A003025, A088957, A133686, A334282, A355529, A355740, A367862, A367867, A367901, A367905, A368094, A368097.

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

from itertools import combinations, product, chain
def v(c):
    for elements in product(*c):
        if len(set(elements)) == len(elements):
            return True
    return False
def a(n):
    if n == 0:
        return 1
    subsets = list(chain.from_iterable(combinations(range(1, n + 1), r) for r in
range(1, n + 1)))
    cs = combinations(subsets, n)
    c = sum(1 for c in cs if v(c))
    return c
[print(a(n)) for n in range(7)] # Robert P. P. McKone, Jan 02 2024

a(n) + A368600(n) = A136556(n).

a(6) from Robert P. P. McKone, Jan 02 2024

a(7)-a(8) from Christian Sievers, Jul 25 2024

A367909 Numbers n such that there is more than one way to choose a different binary index of each binary index of n.

Original entry on oeis.org

4, 12, 16, 18, 20, 32, 33, 36, 48, 52, 64, 65, 66, 68, 72, 76, 80, 82, 84, 96, 97, 100, 112, 132, 140, 144, 146, 148, 160, 161, 164, 176, 180, 192, 193, 194, 196, 200, 204, 208, 210, 212, 224, 225, 228, 240, 256, 258, 260, 264, 266, 268, 272, 274, 276, 288

Offset: 1

Gus Wiseman, Dec 11 2023

nonn

Also BII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice in more than 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.

			The set-system {{1},{1,2},{1,3}} with BII-number 21 satisfies the axiom in only one way (1,2,3), so 21 is not in the sequence.
The terms together with the corresponding set-systems begin:
   4: {{1,2}}
  12: {{1,2},{3}}
  16: {{1,3}}
  18: {{2},{1,3}}
  20: {{1,2},{1,3}}
  32: {{2,3}}
  33: {{1},{2,3}}
  36: {{1,2},{2,3}}
  48: {{1,3},{2,3}}
  52: {{1,2},{1,3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  72: {{3},{1,2,3}}

Wikipedia, Axiom of choice.

These set-systems are counted by A367772.
Positions of terms > 1 in A367905, firsts A367910, sorted firsts A367911.
If there is at least one choice we get A367906, counted by A367902.
If there are no choices we get A367907, counted by A367903.
If there is one unique choice we get A367908, counted by A367904.
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.
A368098 counts unlabeled multiset partitions per axiom, complement A368097.
Cf. A000612, A055621, A072639, A309326, A326702, A326753, A355529, A368100.
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], Length[Select[Tuples[bpe/@bpe[#]], UnsameQ@@#&]]>1&]

A367907 U A367908 U A367909 = A000027.

A367911 Sorted positions of first appearances in A367905.

Original entry on oeis.org

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[[#]]]&]

A370587 Number of subsets of {1..n} containing n such that it is not possible to choose a different prime factor of each element (non-choosable).

Original entry on oeis.org

0, 1, 1, 2, 6, 10, 24, 44, 116, 236, 468, 908, 1960, 3776, 7812, 15876, 32504, 63744, 130104, 257592, 521152, 1042976, 2087096, 4166408, 8376816, 16760832, 33507744, 67089280, 134169440, 268236928, 536759984, 1073233840, 2147384000, 4294503744, 8589075216, 17179048048

Offset: 0

Gus Wiseman, Feb 28 2024

nonn

			The a(0) = 0 through a(5) = 10 subsets:
  .  {1}  {1,2}  {1,3}    {1,4}      {1,5}
                 {1,2,3}  {2,4}      {1,2,5}
                          {1,2,4}    {1,3,5}
                          {1,3,4}    {1,4,5}
                          {2,3,4}    {2,4,5}
                          {1,2,3,4}  {1,2,3,5}
                                     {1,2,4,5}
                                     {1,3,4,5}
                                     {2,3,4,5}
                                     {1,2,3,4,5}

First differences of A370583, complement A370582, cf. A370584.
The complement is counted by A370586.
For a unique choice we have A370588.
For binary indices instead of factors we have A370639, complement A370589.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, indices A112798, length A001222.
A355741 counts choices of a prime factor of each prime index.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
A368098 counts choosable unlabeled multiset partitions, complement A368097.
A368100 ranks choosable multisets, complement A355529.
A368414 counts choosable factorizations, complement A368413.
A370585 counts maximal choosable sets.
A370592 counts choosable partitions, complement A370593.
Cf. A000040, A000720, A005117, A045778, A133686, A355739, A355744, A355745, A367905, A370636.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n] && Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]==0&]],{n,0,10}]

More terms from Jinyuan Wang, Mar 28 2025

A367772 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice in more than one way.

Original entry on oeis.org

0, 0, 1, 23, 1105, 154941, 66072394, 88945612865, 396990456067403

Offset: 0

Gus Wiseman, Dec 12 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.

			Non-isomorphic representatives of the a(3) = 23 set-systems:
  {{1,2}}
  {{1,2,3}}
  {{1},{2,3}}
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}

Wikipedia, Axiom of choice.

For at least one choice we have A367902.
For no choices we have A367903, no singletons A367769, ranks A367907.
For a unique choice we have A367904, ranks A367908.
These set-systems have ranks A367909.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
Cf. A059201, A102896, A133686, A283877, A306445, A323818, A355741, A367770, A367862, A367869, A367901, A367905.

Table[Length[Select[Subsets[Subsets[Range[n]]], Length[Select[Tuples[#], UnsameQ@@#&]]>1&]], {n,0,3}]

A367903(n) + A367904(n) + a(n) = A058891(n).

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

A367910 Least number k such that there are exactly n ways to choose a different binary index of each binary index of k.

Original entry on oeis.org

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

Offset: 0

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:
      7: {{1},{2},{1,2}}
      1: {{1}}
      4: {{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}}
  13376: {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  16704: {{1,2,3},{1,4},{1,2,3,4}}
   5184: {{1,2,3},{1,2,4},{1,3,4}}
  82240: {{1,2,3},{1,4},{1,2,3,4},{1,5}}
  70720: {{1,2,3},{1,2,4},{1,3,4},{1,5}}

Wikipedia, Axiom of choice.

Positions of first appearances in A367905.
The sorted version is A367911.
For multisets w/o distinctness: A367913, firsts of A367912, sorted A367915.
Not requiring distinctness gives A368111, firsts of A368109, sorted A368112.
For multisets of indices we have A368184, firsts of A368183, sorted A368185.
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, A367771, A367906, A367907, A367908, A367909.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
c=Table[Length[Select[Tuples[bpe/@bpe[n]],UnsameQ@@#&]],{n,1000}];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
Table[Position[c,n][[1,1]],{n,0,spnm[c]}]

A368101 Numbers of which there is exactly one way to choose a different prime factor of each prime index.

Original entry on oeis.org

1, 3, 5, 11, 15, 17, 31, 33, 39, 41, 51, 55, 59, 65, 67, 83, 85, 87, 93, 109, 111, 123, 127, 129, 155, 157, 165, 177, 179, 187, 191, 201, 205, 211, 213, 235, 237, 241, 249, 255, 267, 277, 283, 295, 303, 305, 319, 321, 327, 331, 335, 341, 353, 365, 367, 381

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 2795 are {3,6,14}, with prime factors {{3},{2,3},{2,7}}, and the only choice with different terms is {3,2,7}, so 2795 is in the sequence.
The terms together with their prime indices of prime indices begin:
    1: {}
    3: {{1}}
    5: {{2}}
   11: {{3}}
   15: {{1},{2}}
   17: {{4}}
   31: {{5}}
   33: {{1},{3}}
   39: {{1},{1,2}}
   41: {{6}}
   51: {{1},{4}}
   55: {{2},{3}}
   59: {{7}}
   65: {{2},{1,2}}
   67: {{8}}
   83: {{9}}
   85: {{2},{4}}
   87: {{1},{1,3}}
   93: {{1},{5}}
  109: {{10}}
  111: {{1},{1,1,2}}

For no choices we have A355529, odd A355535, binary A367907.
Positions of ones in A367771.
The version for binary indices is A367908, positions of ones in A367905.
For any number of choices we have A368100.
For a unique set instead of sequence we have A370647, counted by A370594.
A058891 counts set-systems, covering A003465, connected A323818.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sort A118914, length A001221, sum A001222.
A355741 chooses a prime factor of each prime index, multisets A355744.
Cf. A007716,  A355737, A355739, A355740, A355745, A367904, A367906, A370584.

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

A368409 Number of non-isomorphic connected set-systems of weight n contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 5, 16, 41, 130

Offset: 0

Gus Wiseman, Dec 25 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(4) = 1 through a(8) = 16 set-systems:
  {1}{2}{12}  .  {1}{2}{13}{23}  {1}{3}{23}{123}    {1}{5}{15}{2345}
                 {1}{2}{3}{123}  {1}{4}{14}{234}    {2}{13}{23}{123}
                 {2}{3}{13}{23}  {2}{3}{23}{123}    {3}{13}{23}{123}
                                 {3}{12}{13}{23}    {3}{4}{34}{1234}
                                 {1}{2}{3}{13}{23}  {1}{2}{13}{24}{34}
                                                    {1}{2}{3}{14}{234}
                                                    {1}{2}{3}{23}{123}
                                                    {1}{2}{3}{4}{1234}
                                                    {1}{3}{4}{14}{234}
                                                    {2}{3}{12}{13}{23}
                                                    {2}{3}{13}{24}{34}
                                                    {2}{3}{14}{24}{34}
                                                    {2}{3}{4}{14}{234}
                                                    {2}{4}{13}{24}{34}
                                                    {3}{4}{13}{24}{34}
                                                    {3}{4}{14}{24}{34}

Wikipedia, Axiom of choice.

For unlabeled graphs we have A140636, connected case of A140637.
For labeled graphs: A140638, connected case of A367867 (complement A133686).
This is the connected case of A368094.
The complement is A368410, connected case of A368095.
Allowing repeats: A368411, connected case of A368097, ranks A355529.
Complement with repeats: A368412, connected case of A368098, ranks A368100.
Allowing repeat edges only: connected case of 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. A134964, A302545, A306005, A317533, A321405, A326031, A367903, A367907, A368187, A368413.

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]}]]];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Union[brute/@Select[mpm[n], UnsameQ@@#&&And@@UnsameQ@@@#&&Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,6}]

A368730 Number of n-element sets of singletons or pairs of distinct elements of {1..n} with union {1..n}, or loop-graphs covering n vertices with n edges, such that it is not possible to choose a different element from each.

Original entry on oeis.org

0, 0, 0, 0, 6, 180, 4560, 117600, 3234588, 96119982, 3092585310, 107542211535, 4029055302855, 162040513972623, 6970457656110039, 319598974394563500, 15568332397812799920, 803271954062642638830, 43778508937914677872788, 2513783434620146896920843

Offset: 0

Gus Wiseman, Jan 04 2024

nonn

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(4) = 6 set-systems:
  {{1},{2},{1,2},{3,4}}
  {{1},{3},{1,3},{2,4}}
  {{1},{4},{1,4},{2,3}}
  {{2},{3},{1,4},{2,3}}
  {{2},{4},{1,3},{2,4}}
  {{3},{4},{1,2},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Graph Loop.
ProofWiki, Definition:Loop-Graph

The case of a unique choice appears to be A000272.
The version without the choice condition is A368597, non-covering A014068.
The complement appears to be A333331.
The non-covering case is A368596, allowing edges of any size A368600.
Allowing any number of edges of any size gives A367903, ranks A367907.
Allowing any number of non-singletons gives A367868, non-covering A367867.
A000085 counts set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
A322661 counts labeled covering half-loop-graphs, connected A062740.
Cf. A000666, A116508, A133686, A367863, A367869, A367901, A368532.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}], {n}],Union@@#==Range[n] && Length[Select[Tuples[#],UnsameQ@@#&]]==0&]],{n,0,5}]

a(n) = A368596(n) + A368597(n) - A014068(n). - Andrew Howroyd, Jan 10 2024

Terms a(7) and beyond from Andrew Howroyd, Jan 10 2024

cp's OEIS Frontend

A368600 Number of ways to choose a set of n nonempty subsets of {1..n} such that it is not possible to choose a different element from each.

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A368601 Number of ways to choose a set of n nonempty subsets of {1..n} such that it is possible to choose a different element from each.

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A367909 Numbers n such that there is more than one way to choose a different binary index of each binary index of n.

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A367911 Sorted positions of first appearances in A367905.

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A370587 Number of subsets of {1..n} containing n such that it is not possible to choose a different prime factor of each element (non-choosable).

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A367772 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice in more than one way.

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A367910 Least number k such that there are exactly n ways to choose a different binary index of each binary index of k.

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A368101 Numbers of which there is exactly one way to choose a different prime factor of each prime index.

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