A368097 -id:A368097 - cp's OEIS frontend

A368421 Number of non-isomorphic set multipartitions of weight n contradicting a strict version of the axiom of choice.

0, 0, 1, 2, 7, 16, 47, 116, 325, 861

Offset: 0

Gus Wiseman, Dec 26 2023

A set multipartition is a finite multiset of finite nonempty sets. The weight of a set multipartition 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 sequence of nonempty sets Y, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 16 set multipartitions:
  {{1},{1}}  {{1},{1},{1}}  {{1},{1},{2,3}}    {{1},{1},{2,3,4}}
             {{1},{2},{2}}  {{1},{2},{1,2}}    {{2},{1,2},{1,2}}
                            {{2},{2},{1,2}}    {{3},{3},{1,2,3}}
                            {{1},{1},{1},{1}}  {{1},{1},{1},{2,3}}
                            {{1},{1},{2},{2}}  {{1},{1},{3},{2,3}}
                            {{1},{2},{2},{2}}  {{1},{2},{2},{1,2}}
                            {{1},{2},{3},{3}}  {{1},{2},{2},{3,4}}
                                               {{1},{2},{3},{2,3}}
                                               {{1},{3},{3},{2,3}}
                                               {{2},{2},{2},{1,2}}
                                               {{1},{1},{1},{1},{1}}
                                               {{1},{1},{2},{2},{2}}
                                               {{1},{2},{2},{2},{2}}
                                               {{1},{2},{2},{3},{3}}
                                               {{1},{2},{3},{3},{3}}
                                               {{1},{2},{3},{4},{4}}

Wikipedia, Axiom of choice.

The case of unlabeled graphs is A140637, complement A134964.
Set multipartitions have ranks A302478, cf. A073576.
The case of labeled graphs is A367867, complement A133686.
With distinct edges we have A368094 connected A368409.
The complement with distinct edges is A368095, connected A368410.
Allowing repeated elements gives A368097, ranks A355529.
The complement allowing repeats is A368098, ranks A368100.
Factorizations of this type are counted by A368413, complement A368414.
The complement is counted by 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, A318360, A367903, A367905, A367907.

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],And@@UnsameQ@@@#&&Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,6}]

A370645 Number of integer factorizations of n into unordered factors > 1 such that only one set can be obtained by choosing a different prime factor of each factor.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 01 2024

nonn

All of these factorizations are co-balanced (A340596).

			The factorization f = (3*6*10) has prime factor choices (3,2,2), (3,3,2), (3,2,5), and (3,3,5), of which only (3,2,5) has all different parts, so f is counted under a(180).
The a(n) factorizations for n = 2, 12, 24, 36, 72, 120, 144, 180, 288:
  (2)  (2*6)  (3*8)   (4*9)   (8*9)   (3*5*8)   (2*72)   (4*5*9)   (3*96)
       (3*4)  (4*6)   (6*6)   (2*36)  (4*5*6)   (3*48)   (5*6*6)   (4*72)
              (2*12)  (2*18)  (3*24)  (2*3*20)  (4*36)   (2*3*30)  (6*48)
                      (3*12)  (4*18)  (2*5*12)  (6*24)   (2*5*18)  (8*36)
                              (6*12)  (2*6*10)  (8*18)   (2*6*15)  (9*32)
                                      (3*4*10)  (9*16)   (2*9*10)  (12*24)
                                                (12*12)  (3*4*15)  (16*18)
                                                         (3*5*12)  (2*144)
                                                         (3*6*10)

Multisets of this type are ranked by A368101, see also A368100, A355529.
For nonexistence we have A368413, complement A368414.
Subsets of this type are counted by A370584, see also A370582, A370583.
Maximal sets of this type are counted by A370585.
The version for partitions is A370594, see also A370592, A370593.
Subsets of this type are counted by A370638, see also A370636, A370637.
For unlabeled multiset partitions we have A370646, also A368098, A368097.
A001055 counts factorizations, strict A045778.
A006530 gives greatest prime factor, least A020639.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A027746 lists prime factors, A112798 indices, length A001222.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
A355741 counts ways to choose a prime factor of each prime index.
For set-systems see A367902-A367908.
Cf. A000040, A000720, A003963, A340596, A340653, A355744, A355745, A368110.

facs[n_]:=If[n<=1,{{}},Join @@ Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Length[Union[Sort/@Select[Tuples[First /@ FactorInteger[#]&/@#], UnsameQ@@#&]]]==1&]],{n,100}]

A368412 Number of non-isomorphic connected multiset partitions of weight n satisfying a strict version of the axiom of choice.

Original entry on oeis.org

0, 1, 2, 4, 11, 25, 75, 206, 650, 2049, 6895

Offset: 0

Gus Wiseman, Dec 26 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(1) = 1 through a(4) = 11 multiset partitions:
  {{1}}  {{1,1}}  {{1,1,1}}    {{1,1,1,1}}
         {{1,2}}  {{1,2,2}}    {{1,1,2,2}}
                  {{1,2,3}}    {{1,2,2,2}}
                  {{2},{1,2}}  {{1,2,3,3}}
                               {{1,2,3,4}}
                               {{1},{1,2,2}}
                               {{1,2},{1,2}}
                               {{1,2},{2,2}}
                               {{1,3},{2,3}}
                               {{2},{1,2,2}}
                               {{3},{1,2,3}}

Wikipedia, Axiom of choice.

The case of labeled graphs is A129271, connected case of A133686.
The complement for labeled graphs is A140638, connected case of A367867.
This is the connected case of A368098, ranks A368100.
Complement set-systems: A368409, connected case of A368094, ranks A367907.
For set-systems we have A368410, connected case of A368095, ranks A367906.
The complement is A368411, connected case of A368097, ranks A355529.
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. A140637, A302545, A316983, A317533, A319616, A367902, A367905, A368187.

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],Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]!={}&]]],{n,0,6}]

A368411 Number of non-isomorphic connected multiset partitions of weight n contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 1, 2, 6, 15, 50, 148, 509, 1725, 6218

Offset: 0

Gus Wiseman, Dec 26 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(2) = 1 through a(5) = 15 multiset partitions:
  {{1},{1}}  {{1},{1,1}}    {{1},{1,1,1}}      {{1},{1,1,1,1}}
             {{1},{1},{1}}  {{1,1},{1,1}}      {{1,1},{1,1,1}}
                            {{1},{1},{1,1}}    {{1},{1},{1,1,1}}
                            {{1},{2},{1,2}}    {{1},{1,1},{1,1}}
                            {{2},{2},{1,2}}    {{1},{1},{1,2,2}}
                            {{1},{1},{1},{1}}  {{1},{1,2},{2,2}}
                                               {{1},{2},{1,2,2}}
                                               {{2},{1,2},{1,2}}
                                               {{2},{1,2},{2,2}}
                                               {{2},{2},{1,2,2}}
                                               {{3},{3},{1,2,3}}
                                               {{1},{1},{1},{1,1}}
                                               {{1},{2},{2},{1,2}}
                                               {{2},{2},{2},{1,2}}
                                               {{1},{1},{1},{1},{1}}

Wikipedia, Axiom of choice.

The case of labeled graphs is A140638, connected case of A367867.
The complement for labeled graphs is A129271, connected case of A133686.
This is the connected case of A368097.
For set-systems we have A368409, connected case of A368094, ranks A367907.
Complement set-systems: A368410, connected case of A368095, ranks A367906.
The complement is A368412, connected case of A368098, ranks A368100.
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. A140637, A302545, A316983, A317533, A319616, A367903, A367905, A368187.

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],Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,6}]

A370591 Number of minimal subsets of {1..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, 1, 2, 2, 4, 4, 7, 11, 16, 16, 30, 30, 39, 73

Offset: 0

Gus Wiseman, Feb 28 2024

			The a(1) = 1 through a(10) = 16 subsets:
{1}  {1}  {1}  {1}    {1}    {1}      {1}      {1}      {1}      {1}
               {2,4}  {2,4}  {2,4}    {2,4}    {2,4}    {2,4}    {2,4}
                             {2,3,6}  {2,3,6}  {2,8}    {2,8}    {2,8}
                             {3,4,6}  {3,4,6}  {4,8}    {3,9}    {3,9}
                                               {2,3,6}  {4,8}    {4,8}
                                               {3,4,6}  {2,3,6}  {2,3,6}
                                               {3,6,8}  {2,6,9}  {2,6,9}
                                                        {3,4,6}  {3,4,6}
                                                        {3,6,8}  {3,6,8}
                                                        {4,6,9}  {4,6,9}
                                                        {6,8,9}  {6,8,9}
                                                                 {2,5,10}
                                                                 {4,5,10}
                                                                 {5,8,10}
                                                                 {3,5,6,10}
                                                                 {5,6,9,10}

Minimal case of A370583, complement A370582.
For binary indices instead of factors we have A370642, minima of A370637.
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, A045778, A133686, A355739, A355744, A355745, A367771, A370584, A370586, A370587, A370589.

Table[Length[fasmin[Select[Subsets[Range[n]], Length[Select[Tuples[prix/@#],UnsameQ@@#&]]==0&]]], {n,0,15}]

A370588 Number of subsets of {1..n} containing n such that only one set can be obtained by choosing a different prime factor of each element.

Original entry on oeis.org

0, 0, 1, 2, 2, 6, 6, 18, 12, 20, 36, 104, 76, 284, 320, 408, 252, 1548, 872, 3968, 2800, 4704, 8568, 24008, 10832, 14832, 40688, 18240, 43632, 176240, 97344, 449824, 95328, 404992, 760752, 698864, 436464, 3296048, 3564576, 4057904, 2677776, 16892352, 8676576

Offset: 0

Gus Wiseman, Feb 28 2024

nonn

For example, the only choice of a different prime factor of each element of (4,5,6) is (2,5,3), so {4,5,6} is counted under a(6).

			The a(0) = 0 through a(8) = 12 subsets:
  .  .  {2}  {3}    {4}    {5}      {2,6}    {7}        {8}
             {2,3}  {3,4}  {2,5}    {3,6}    {2,7}      {3,8}
                           {3,5}    {4,6}    {3,7}      {5,8}
                           {4,5}    {2,5,6}  {4,7}      {6,8}
                           {2,3,5}  {3,5,6}  {5,7}      {7,8}
                           {3,4,5}  {4,5,6}  {2,3,7}    {3,5,8}
                                             {2,5,7}    {3,7,8}
                                             {2,6,7}    {5,6,8}
                                             {3,4,7}    {5,7,8}
                                             {3,5,7}    {6,7,8}
                                             {3,6,7}    {3,5,7,8}
                                             {4,5,7}    {5,6,7,8}
                                             {4,6,7}
                                             {2,3,5,7}
                                             {2,5,6,7}
                                             {3,4,5,7}
                                             {3,5,6,7}
                                             {4,5,6,7}

First differences of A370584, cf. A370582, complement A370583.
For any number of choices we have A370586, complement A370587.
For binary indices see A370638, 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.
A370636 counts choosable subsets for binary indices, complement A370637.
Cf. A000040, A000720, A005117, A045778, A133686, A355739, A355744, A355745, A367771, A367905.

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

More terms from Jinyuan Wang, Mar 28 2025

A370590 Number of maximal subsets of {1..n} containing n such that it is possible to choose a different prime factor of each element (choosable).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 5, 2, 4, 14, 25, 13, 38, 46, 66, 28, 178, 57, 235, 106, 238, 656, 1235, 288, 445, 2192, 664, 2016, 6840, 2300, 9140, 888, 6236, 17692, 14724, 7320, 56000, 60472, 70252, 37160, 223884, 66428, 290312, 113172, 80544, 517392, 1001420, 114336

Offset: 0

Gus Wiseman, Feb 28 2024

nonn

For example, the set {4,7,9,10} has choice (2,7,3,5) so is counted under a(10).

			The a(0) = 0 through a(10) = 14 subsets (A = 10):
  .  .  2  23  34  235  256  2357  3578  2579  237A
                   345  356  2567  5678  4579  267A
                        456  3457        5679  279A
                             3567        5789  347A
                             4567              357A
                                               367A
                                               378A
                                               467A
                                               479A
                                               567A
                                               579A
                                               678A
                                               679A
                                               789A

Not requiring n gives A370585, maximal case of A370582, complement A370583.
Maximal case of A370586, complement A370587, unique A370588.
An opposite version is A370591.
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.
A370592 counts choosable partitions, complement A370593.
Cf. A000040, A000720, A133686, A355739, A355744, A355745, A367771, A370584, A370636, A370639.

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

More terms from Jinyuan Wang, Feb 14 2025

A368532 Minimal numbers whose binary indices of binary indices contradict a strict version of the axiom of choice.

Original entry on oeis.org

7, 25, 30, 42, 45, 51, 53, 54, 60, 75, 77, 78, 83, 85, 86, 90, 92, 99, 101, 102, 105, 108, 113, 114, 116, 120, 385, 390, 408, 428, 434, 436, 458, 460, 466, 468, 482, 484, 488, 496, 642, 645, 668, 680, 689, 692, 713, 716, 721, 724, 728, 737, 740, 752, 771, 773

Offset: 1

Gus Wiseman, Dec 29 2023

nonn

Minimality is relative to the ordering where x < y means the binary indices of x are a subset of those of y (a Boolean algebra).
A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion.
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 terms the corresponding set-systems begin:
   7: {{1},{2},{1,2}}
  25: {{1},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  42: {{2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  51: {{1},{2},{1,3},{2,3}}
  53: {{1},{1,2},{1,3},{2,3}}
  54: {{2},{1,2},{1,3},{2,3}}
  60: {{1,2},{3},{1,3},{2,3}}
  75: {{1},{2},{3},{1,2,3}}
  77: {{1},{1,2},{3},{1,2,3}}
  78: {{2},{1,2},{3},{1,2,3}}
  83: {{1},{2},{1,3},{1,2,3}}
  85: {{1},{1,2},{1,3},{1,2,3}}
  86: {{2},{1,2},{1,3},{1,2,3}}
  90: {{2},{3},{1,3},{1,2,3}}
  92: {{1,2},{3},{1,3},{1,2,3}}
  99: {{1},{2},{2,3},{1,2,3}}

The version for MM-numbers of multiset partitions is A368187.
A000110 counts set partitions.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A007716, A134964, A355529, A367905, A367907.
Cf. A140637, A367867, A367903, A368094, A368097, A368413.

vmin[y_]:=Select[y,Function[s,Select[DeleteCases[y,s], SubsetQ[bpe[s],bpe[#]]&]=={}]];
Select[Range[100],Select[Tuples[bpe/@bpe[#]] ,UnsameQ@@#&]=={}&]//vmin

A370646 Number of non-isomorphic multiset partitions of weight n such that only one set can be obtained by choosing a different element of each block.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 62, 165, 475, 1400, 4334

Offset: 0

Gus Wiseman, Mar 12 2024

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.

			The multiset partition {{3},{1,3},{2,3}} has unique choice (3,1,2) so is counted under a(5).
Representatives of the a(1) = 1 through a(5) = 23 multiset partitions:
  {1}  {11}    {111}      {1111}        {11111}
       {1}{2}  {1}{22}    {1}{122}      {11}{122}
               {2}{12}    {11}{22}      {1}{1222}
               {1}{2}{3}  {12}{12}      {11}{222}
                          {1}{222}      {12}{122}
                          {12}{22}      {1}{2222}
                          {2}{122}      {12}{222}
                          {1}{2}{33}    {2}{1122}
                          {1}{3}{23}    {2}{1222}
                          {1}{2}{3}{4}  {22}{122}
                                        {1}{2}{233}
                                        {1}{22}{33}
                                        {1}{23}{23}
                                        {1}{2}{333}
                                        {1}{23}{33}
                                        {1}{3}{233}
                                        {2}{12}{33}
                                        {2}{13}{23}
                                        {2}{3}{123}
                                        {3}{13}{23}
                                        {1}{2}{3}{44}
                                        {1}{2}{4}{34}
                                        {1}{2}{3}{4}{5}

For existence we have A368098, complement A368097.
Multisets of this type are ranked by A368101, see also A368100, A355529.
Subsets of this type are counted by A370584, see also A370582, A370583.
Maximal sets of this type are counted by A370585.
Partitions of this type are counted by A370594, see also A370592, A370593.
Subsets of this type are also counted by A370638, see also A370636, A370637.
Factorizations of this type are A370645, see also A368414, A368413.
Set-systems of this type are A370818, see also A367902, A367903.
A000110 counts set partitions, non-isomorphic A000041.
A001055 counts factorizations, strict A045778.
A007716 counts non-isomorphic multiset partitions, connected A007718.
Cf. A000612, A055621, A283877, A300913, A302545, A316983, A319616, A330223, A368095, A368412, A368422.

cp's OEIS Frontend

A368421 Number of non-isomorphic set multipartitions of weight n contradicting a strict version of the axiom of choice.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A370645 Number of integer factorizations of n into unordered factors > 1 such that only one set can be obtained by choosing a different prime factor of each factor.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A368412 Number of non-isomorphic connected multiset partitions of weight n satisfying a strict version of the axiom of choice.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A368411 Number of non-isomorphic connected multiset partitions of weight n contradicting a strict version of the axiom of choice.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A370591 Number of minimal subsets of {1..n} such that it is not possible to choose a different prime factor of each element (non-choosable).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A370588 Number of subsets of {1..n} containing n such that only one set can be obtained by choosing a different prime factor of each element.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A370590 Number of maximal subsets of {1..n} containing n such that it is possible to choose a different prime factor of each element (choosable).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A368532 Minimal numbers whose binary indices of binary indices contradict a strict version of the axiom of choice.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A370646 Number of non-isomorphic multiset partitions of weight n such that only one set can be obtained by choosing a different element of each block.

Views

Author