A330223 -id:A330223 - cp's OEIS frontend

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

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}]

A330098 Number of distinct multisets of multisets that can be obtained by permuting the vertices of the multiset of multisets with MM-number n.

Original entry on oeis.org

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, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Dec 09 2019

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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

a(n) is a divisor of A303975(n)!.

			The vertex-permutations of {{1,2},{2,3,3}} are:
  {{1,2},{1,3,3}}
  {{1,2},{2,3,3}}
  {{1,3},{1,2,2}}
  {{1,3},{2,2,3}}
  {{2,3},{1,1,2}}
  {{2,3},{1,1,3}}
so a(4927) = 6.

Positions of 1's are A330232.
Positions of first appearances are A330230 and A330233.
The BII-number version is A330231.
Cf. A001055, A003238, A007716, A055621, A056239, A112798, A302242, A303975, A322847, A330194, A330218, A330223, A330227, A330236.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[graprms[primeMS/@primeMS[n]]],{n,100}]

A368095 Number of non-isomorphic set-systems of weight n satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 39, 86, 208, 508, 1304

Offset: 0

Gus Wiseman, Dec 24 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.

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(5) = 17 set-systems:
  {1}  {12}    {123}      {1234}        {12345}
       {1}{2}  {1}{23}    {1}{234}      {1}{2345}
               {2}{12}    {12}{34}      {12}{345}
               {1}{2}{3}  {13}{23}      {14}{234}
                          {3}{123}      {23}{123}
                          {1}{2}{34}    {4}{1234}
                          {1}{3}{23}    {1}{2}{345}
                          {1}{2}{3}{4}  {1}{23}{45}
                                        {1}{24}{34}
                                        {1}{4}{234}
                                        {2}{13}{23}
                                        {2}{3}{123}
                                        {3}{13}{23}
                                        {4}{12}{34}
                                        {1}{2}{3}{45}
                                        {1}{2}{4}{34}
                                        {1}{2}{3}{4}{5}

For labeled graphs we have A133686, complement A367867.
For unlabeled graphs we have A134964, complement A140637.
For set-systems we have A367902, complement A367903.
These set-systems have BII-numbers A367906, complement A367907.
The complement is A368094, connected A368409.
Repeats allowed: A368098, ranks A368100, complement A368097, ranks A355529.
Minimal multiset partitions not of this type are counted by A368187.
The connected case is A368410.
Factorizations of this type are counted by A368414, complement A368413.
Allowing repeated edges gives A368422, complement A368421.
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. A001055, A007717, A302545, A306005, A316983, A319560, A321194, A321405, A330223, A367769, A367901.

Table[Length[Select[bmp[n], UnsameQ@@#&&And@@UnsameQ@@@#&&Select[Tuples[#], UnsameQ@@#&]!={}&]], {n,0,10}]

A368098 Number of non-isomorphic multiset partitions of weight n satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 1, 3, 7, 21, 54, 165, 477, 1501, 4736, 15652

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(1) = 1 through a(4) = 21 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1,2}}    {{1,2,2}}      {{1,1,2,2}}
         {{1},{2}}  {{1,2,3}}      {{1,2,2,2}}
                    {{1},{2,2}}    {{1,2,3,3}}
                    {{1},{2,3}}    {{1,2,3,4}}
                    {{2},{1,2}}    {{1},{1,2,2}}
                    {{1},{2},{3}}  {{1,1},{2,2}}
                                   {{1,2},{1,2}}
                                   {{1},{2,2,2}}
                                   {{1,2},{2,2}}
                                   {{1},{2,3,3}}
                                   {{1,2},{3,3}}
                                   {{1},{2,3,4}}
                                   {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{2},{3,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{1},{2},{3},{4}}

Wikipedia, Axiom of choice.

The case of labeled graphs is A133686, complement A367867.
The case of unlabeled graphs is A134964, complement A140637 (apparently).
Set-systems of this type are A367902, ranks A367906, connected A368410.
The complimentary set-systems are A367903, ranks A367907, connected A368409.
For set-systems we have A368095, complement A368094.
The complement is A368097, ranks A355529.
These multiset partitions have ranks A368100.
The connected case is A368412, complement A368411.
Factorizations of this type are counted by A368414, complement A368413.
For set multipartitions we have A368422, complement A368421.
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, A330223, A330227, 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]}]]];
Table[Length[Union[brute/@Select[mpm[n], Select[Tuples[#],UnsameQ@@#&]!={}&]]], {n,0,6}]

A330227 Number of non-isomorphic fully chiral multiset partitions of weight n.

Original entry on oeis.org

1, 1, 2, 7, 16, 49, 144, 447, 1417, 4707

Offset: 0

Gus Wiseman, Dec 08 2019

A multiset partition is fully chiral if every permutation of the vertices gives a different representative. The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 16 multiset partitions:
  {1}  {11}    {111}      {1111}
       {1}{1}  {122}      {1222}
               {1}{11}    {1}{111}
               {1}{22}    {11}{11}
               {2}{12}    {1}{122}
               {1}{1}{1}  {1}{222}
               {1}{2}{2}  {12}{22}
                          {1}{233}
                          {2}{122}
                          {1}{1}{11}
                          {1}{1}{22}
                          {1}{2}{22}
                          {1}{3}{23}
                          {2}{2}{12}
                          {1}{1}{1}{1}
                          {1}{2}{2}{2}

MM-numbers of these multiset partitions are the odd terms of A330236.
Non-isomorphic costrict (or T_0) multiset partitions are A316980.
Non-isomorphic achiral multiset partitions are A330223.
BII-numbers of fully chiral set-systems are A330226.
Fully chiral partitions are counted by A330228.
Fully chiral covering set-systems are A330229.
Fully chiral factorizations are A330235.
Cf. A000612, A001055, A007716, A055621, A283877, A317533, A322847, A330098, A330232.

A330232 MM-numbers of achiral multisets of multisets.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 72, 73, 76, 79, 80

Offset: 1

Gus Wiseman, Dec 08 2019

nonn

First differs from A322554 in lacking 141.
A multiset of multisets is achiral if it is not changed by any permutation of the vertices.
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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of non-achiral multisets of multisets (the complement of this sequence) together with their MM-numbers begins:
  35: {{2},{1,1}}
  37: {{1,1,2}}
  39: {{1},{1,2}}
  45: {{1},{1},{2}}
  61: {{1,2,2}}
  65: {{2},{1,2}}
  69: {{1},{2,2}}
  70: {{},{2},{1,1}}
  71: {{1,1,3}}
  74: {{},{1,1,2}}
  75: {{1},{2},{2}}
  77: {{1,1},{3}}
  78: {{},{1},{1,2}}
  87: {{1},{1,3}}
  89: {{1,1,1,2}}
  90: {{},{1},{1},{2}}

The fully-chiral version is A330236.
Achiral set-systems are counted by A083323.
MG-numbers of planted achiral trees are A214577.
MM-weight is A302242.
MM-numbers of costrict (or T_0) multisets of multisets are A322847.
BII-numbers of achiral set-systems are A330217.
Non-isomorphic achiral multiset partitions are A330223.
Achiral integer partitions are counted by A330224.
Achiral factorizations are A330234.
Cf. A001055, A003238, A007716, A056239, A112798, A303975, A330098, A330230, A330233.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Apply[Rule,Table[{p[[i]],i},{i,Length[p]}],{1}])],{p,Permutations[Union@@m]}]]
Select[Range[100],Length[graprms[primeMS/@primeMS[#]]]==1&]

A330236 MM-numbers of fully chiral multisets of multisets.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 53, 54, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Dec 10 2019

nonn

A multiset of multisets is fully chiral every permutation of the vertices gives a different representative.

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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of all fully chiral multisets of multisets together with their MM-numbers begins:
   1:             18: {}{1}{1}      37: {112}          57: {1}{111}
   2: {}          19: {111}         38: {}{111}        59: {7}
   3: {1}         20: {}{}{2}       39: {1}{12}        61: {122}
   4: {}{}        21: {1}{11}       40: {}{}{}{2}      62: {}{5}
   5: {2}         22: {}{3}         41: {6}            63: {1}{1}{11}
   6: {}{1}       23: {22}          42: {}{1}{11}      64: {}{}{}{}{}{}
   7: {11}        24: {}{}{}{1}     44: {}{}{3}        65: {2}{12}
   8: {}{}{}      25: {2}{2}        45: {1}{1}{2}      67: {8}
   9: {1}{1}      27: {1}{1}{1}     46: {}{22}         68: {}{}{4}
  10: {}{2}       28: {}{}{11}      48: {}{}{}{}{1}    69: {1}{22}
  11: {3}         31: {5}           49: {11}{11}       70: {}{2}{11}
  12: {}{}{1}     32: {}{}{}{}{}    50: {}{2}{2}       71: {113}
  14: {}{11}      34: {}{4}         53: {1111}         72: {}{}{}{1}{1}
  16: {}{}{}{}    35: {2}{11}       54: {}{1}{1}{1}    74: {}{112}
  17: {4}         36: {}{}{1}{1}    56: {}{}{}{11}     75: {1}{2}{2}
The complement starts: {13, 15, 26, 29, 30, 33, 43, 47, 51, 52, 55, 58, 60, 66, 73, 79, 85, 86, 93, 94}.

Costrict (or T_0) factorizations are A316978.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Full chiral partitions are A330228.
Fully chiral covering set-systems are A330229.
Fully chiral factorizations are A330235.
Cf. A001055, A007716, A083323, A112798, A303975, A317533, A330098, A330223, A330224, A330232.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Select[Range[100],Length[graprms[primeMS/@primeMS[#]]]==Length[Union@@primeMS/@primeMS[#]]!&]

Numbers n such that A330098(n) = A303975(n)!.

A330217 BII-numbers of achiral set-systems.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 9, 10, 11, 16, 25, 32, 42, 52, 63, 64, 75, 116, 127, 128, 129, 130, 131, 136, 137, 138, 139, 256, 385, 512, 642, 772, 903, 1024, 1155, 1796, 1927, 2048, 2184, 2320, 2457, 2592, 2730, 2868, 3007, 4096, 4233, 6416, 6553, 8192, 8330

Offset: 1

Gus Wiseman, Dec 06 2019

nonn

A set-system is a finite set of finite nonempty sets. It is achiral if it is not changed by any permutation of the vertices.

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system has a different BII-number. For example, 18 has reversed binary expansion (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. Elements of a set-system are sometimes called edges.

			The sequence of all achiral set-systems together with their BII-numbers begins:
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   4: {{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  16: {{1,3}}
  25: {{1},{3},{1,3}}
  32: {{2,3}}
  42: {{2},{3},{2,3}}
  52: {{1,2},{1,3},{2,3}}
  63: {{1},{2},{3},{1,2},{1,3},{2,3}}
  64: {{1,2,3}}
  75: {{1},{2},{3},{1,2,3}}

These are numbers n such that A330231(n) = 1.
Achiral set-systems are counted by A083323.
MG-numbers of planted achiral trees are A214577.
Non-isomorphic achiral multiset partitions are A330223.
Achiral integer partitions are counted by A330224.
BII-numbers of fully chiral set-systems are A330226.
MM-numbers of achiral multisets of multisets are A330232.
Achiral factorizations are A330234.
Cf. A000120, A003238, A016031, A048793, A070939, A326031, A326702, A327080, A327081, A330218, A330229, A330233.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Select[Range[0,1000],Length[graprms[bpe/@bpe[#]]]==1&]

A330230 Least MM-number of a multiset of multisets with n distinct representatives obtainable by permuting the vertices.

Original entry on oeis.org

1, 35, 141, 1713, 28011, 355

Offset: 1

Gus Wiseman, Dec 09 2019

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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their corresponding multisets of multisets begins:
      1: {}
     35: {{2},{1,1}}
    141: {{1},{2,3}}
   1713: {{1},{2,3,4}}
  28011: {{1},{2,3,4,5}}
    355: {{2},{1,1,3}}

The BII-number version is A330218.
Positions of first appearances in A330098.
The sorted version is A330233.
MM-numbers of achiral multisets of multisets are A330232.
MM-numbers of fully-chiral multisets of multisets are A330236.
Cf. A001055, A003238, A007716, A056239, A112798, A302242, A303975, A322847, A330103, A330223, A330227, A330231, A330236.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Apply[Rule,Table[{p[[i]],i},{i,Length[p]}],{1}])],{p,Permutations[Union@@m]}]];
dv=Table[Length[graprms[primeMS/@primeMS[n]]],{n,1000}];
Table[Position[dv,i][[1,1]],{i,First[Split[Union[dv],#1+1==#2&]]}]

A330228 Number of fully chiral integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 12, 18, 25, 33, 45, 61, 80, 106, 140, 176, 232, 293, 381, 476, 615, 764, 975, 1191, 1511, 1849, 2322, 2812, 3517, 4231, 5240, 6297, 7736, 9260, 11315, 13468, 16378, 19485, 23531, 27851, 33525, 39585, 47389, 55844, 66517, 78169, 92810

Offset: 0

Gus Wiseman, Dec 08 2019

nonn

A multiset partition is fully chiral if every permutation of the vertices gives a different representative. An integer partition is fully chiral if taking the multiset of prime indices of each part gives a fully chiral multiset of multisets.

			The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (33)      (7)
       (11)  (21)   (22)    (41)     (42)      (43)
             (111)  (31)    (221)    (51)      (322)
                    (211)   (311)    (222)     (331)
                    (1111)  (2111)   (411)     (421)
                            (11111)  (2211)    (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

The Heinz numbers of these partitions are given by A330236.
Costrict (or T_0) partitions are A319564.
Achiral partitions are A330224.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic, fully chiral multiset partitions are A330227.
Fully chiral covering set-systems are A330229.
Fully chiral factorizations are A330235.
Cf. A001055, A007716, A322847, A330098, A330223.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[IntegerPartitions[n],Length[graprms[primeMS/@#]]==Length[Union@@primeMS/@#]!&]],{n,0,15}]

cp's OEIS Frontend

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

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A330098 Number of distinct multisets of multisets that can be obtained by permuting the vertices of the multiset of multisets with MM-number n.

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A368095 Number of non-isomorphic set-systems of weight n satisfying a strict version of the axiom of choice.

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A368098 Number of non-isomorphic multiset partitions of weight n satisfying a strict version of the axiom of choice.

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A330227 Number of non-isomorphic fully chiral multiset partitions of weight n.

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A330232 MM-numbers of achiral multisets of multisets.

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A330236 MM-numbers of fully chiral multisets of multisets.

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Formula

A330217 BII-numbers of achiral set-systems.

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A330230 Least MM-number of a multiset of multisets with n distinct representatives obtainable by permuting the vertices.

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