A302545 -id:A302545 - cp's OEIS frontend

A321413 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons and relatively prime part sizes.

1, 0, 0, 0, 0, 3, 0, 14, 13, 50, 65

Offset: 0

Gus Wiseman, Nov 16 2018

Also the number of nonnegative integer symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with relatively prime row sums (or column sums) and no row (or column) summing to 1.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
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(5) = 3, a(7) = 14, and a(8) = 13 multiset partitions:
  {{11}{122}}  {{111}{1222}}    {{111}{11222}}
  {{11}{222}}  {{111}{2222}}    {{111}{22222}}
  {{12}{122}}  {{112}{1222}}    {{112}{12222}}
               {{11}{22222}}    {{122}{11222}}
               {{12}{12222}}    {{11}{122}{233}}
               {{122}{1122}}    {{11}{122}{333}}
               {{22}{11222}}    {{11}{222}{333}}
               {{11}{12}{233}}  {{11}{223}{233}}
               {{11}{22}{233}}  {{12}{122}{333}}
               {{11}{22}{333}}  {{12}{123}{233}}
               {{11}{23}{233}}  {{13}{112}{233}}
               {{12}{12}{333}}  {{13}{122}{233}}
               {{12}{13}{233}}  {{23}{123}{123}}
               {{13}{23}{123}}

Cf. A000219, A007716, A120733, A138178, A302545, A316983, A319616.

Cf. A320796, A320797, A320806, A320813, A321283, A321409, A321410, A321411.

A321677 Number of non-isomorphic set multipartitions (multisets of sets) of weight n with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 4, 4, 16, 22, 70, 132, 375, 848, 2428, 6256, 18333, 52560, 161436, 500887, 1624969, 5384625, 18438815, 64674095, 233062429, 859831186, 3248411250, 12545820860, 49508089411, 199410275018, 819269777688, 3430680180687, 14633035575435, 63535672197070

Offset: 0

Gus Wiseman, Nov 16 2018

nonn

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(2) = 1 through a(6) = 16 set multipartitions:
  {{1,2}}  {{1,2,3}}  {{1,2,3,4}}    {{1,2,3,4,5}}    {{1,2,3,4,5,6}}
                      {{1,2},{1,2}}  {{1,2},{3,4,5}}  {{1,2,3},{1,2,3}}
                      {{1,2},{3,4}}  {{1,4},{2,3,4}}  {{1,2},{3,4,5,6}}
                      {{1,3},{2,3}}  {{2,3},{1,2,3}}  {{1,2,3},{4,5,6}}
                                                      {{1,2,5},{3,4,5}}
                                                      {{1,3,4},{2,3,4}}
                                                      {{1,5},{2,3,4,5}}
                                                      {{3,4},{1,2,3,4}}
                                                      {{1,2},{1,2},{1,2}}
                                                      {{1,2},{1,3},{2,3}}
                                                      {{1,2},{3,4},{3,4}}
                                                      {{1,2},{3,4},{5,6}}
                                                      {{1,2},{3,5},{4,5}}
                                                      {{1,3},{2,3},{2,3}}
                                                      {{1,3},{2,4},{3,4}}
                                                      {{1,4},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000219, A007716, A049311, A283877, A302545, A316983, A319616.

Cf. A320797, A320798, A320804, A320811, A320812, A321404, A321406.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)) - Vec(sum(j=1, #q, if(t%q[j]==0, q[j])) + O(x*x^k), -k)}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, n, subst(x*Ser(K(q, t, n\t)/t),x,x^t) )); s+=permcount(q)*polcoef(exp(g), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2024

Terms a(11) and beyond from Andrew Howroyd, Sep 01 2019

A330124 Number of unlabeled set-systems with n vertices and no endpoints.

Original entry on oeis.org

1, 1, 2, 22, 1776

Offset: 0

Gus Wiseman, Dec 05 2019

A set-system is a finite set of finite nonempty sets. An endpoint is a vertex appearing only once (degree 1).

			Non-isomorphic representatives of the a(3) = 22 set-systems:
  0
  {1}{2}{12}
  {12}{13}{23}
  {1}{23}{123}
  {12}{13}{123}
  {1}{2}{13}{23}
  {1}{2}{3}{123}
  {1}{12}{13}{23}
  {1}{2}{13}{123}
  {1}{12}{13}{123}
  {1}{12}{23}{123}
  {12}{13}{23}{123}
  {1}{2}{3}{12}{13}
  {1}{2}{12}{13}{23}
  {1}{2}{3}{12}{123}
  {1}{2}{12}{13}{123}
  {1}{2}{13}{23}{123}
  {1}{12}{13}{23}{123}
  {1}{2}{3}{12}{13}{23}
  {1}{2}{3}{12}{13}{123}
  {1}{2}{12}{13}{23}{123}
  {1}{2}{3}{12}{13}{23}{123}

Partial sums of the covering case A330196.
The labeled version is A330059.
The "multi" version is A302545.
Unlabeled set-systems with no endpoints counted by weight are A330054.
Unlabeled set-systems with no singletons are A317794.
Unlabeled set-systems counted by vertices are A000612.
Unlabeled set-systems counted by weight are A283877.
The case with no singletons is A320665.
Cf. A007716, A016031, A306005, A317795, A321405, A330052, A330055.

A368096 Triangle read by rows where T(n,k) is the number of non-isomorphic set-systems of length k and weight n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 5, 8, 3, 1, 0, 1, 8, 18, 13, 3, 1, 0, 1, 9, 32, 37, 15, 3, 1, 0, 1, 13, 55, 96, 59, 16, 3, 1, 0, 1, 14, 91, 209, 196, 74, 16, 3, 1, 0, 1, 19, 138, 449, 573, 313, 82, 16, 3, 1, 0, 1, 20, 206, 863, 1529, 1147, 403, 84, 16, 3, 1

Offset: 0

Gus Wiseman, Dec 28 2023

A set-system is a finite set of finite nonempty sets.

Conjecture: Column k = 2 is A101881.

			Triangle begins:
   1
   0   1
   0   1   1
   0   1   2   1
   0   1   4   3   1
   0   1   5   8   3   1
   0   1   8  18  13   3   1
   0   1   9  32  37  15   3   1
   0   1  13  55  96  59  16   3   1
   0   1  14  91 209 196  74  16   3   1
   0   1  19 138 449 573 313  82  16   3   1
   ...
Non-isomorphic representatives of the set-systems counted in row n = 5:
  .  {12345}  {1}{1234}  {1}{2}{123}  {1}{2}{3}{12}  {1}{2}{3}{4}{5}
              {1}{2345}  {1}{2}{134}  {1}{2}{3}{14}
              {12}{123}  {1}{2}{345}  {1}{2}{3}{45}
              {12}{134}  {1}{12}{13}
              {12}{345}  {1}{12}{23}
                         {1}{12}{34}
                         {1}{23}{24}
                         {1}{23}{45}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A283877, connected case A300913.
For multiset partitions we have A317533.
Counting connected components instead of edges gives A321194.
For set multipartitions we have A334550.
For strict multiset partitions we have A368099.
A000110 counts set-partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A049311 counts non-isomorphic set multipartitions, connected A056156.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A316980 counts non-isomorphic strict multiset partitions, connected A319557.
A319559 counts non-isomorphic T_0 set-systems, connected A319566.
Cf. A101881, A255903, A302545, A306005, A317532, A317794, A317795, A319560, A321405, A368094, A368095.

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@@@#&&Length[#]==k&]]], {n,0,5},{k,0,n}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
G(n)={my(s=0); forpart(q=n, my(p=sum(t=1, n, y^t*subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*exp(p-subst(subst(p, x, x^2), y, y^2))); s/n!}
T(n)={[Vecrev(p) | p <- Vec(G(n))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

Terms a(66) and beyond from Andrew Howroyd, Jan 11 2024

A368099 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element sets of finite nonempty multisets with cardinalities summing to n, or strict multiset partitions of weight n and length k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 12, 5, 1, 0, 7, 28, 22, 5, 1, 0, 11, 66, 83, 31, 5, 1, 0, 15, 134, 252, 147, 34, 5, 1, 0, 22, 280, 726, 620, 203, 35, 5, 1, 0, 30, 536, 1946, 2283, 1069, 235, 35, 5, 1, 0, 42, 1043, 4982, 7890, 5019, 1469, 248, 35, 5, 1

Offset: 0

Gus Wiseman, Dec 31 2023

			Triangle begins:
    1
    0    1
    0    2    1
    0    3    4    1
    0    5   12    5    1
    0    7   28   22    5    1
    0   11   66   83   31    5    1
    0   15  134  252  147   34    5    1
    0   22  280  726  620  203   35    5    1
    0   30  536 1946 2283 1069  235   35    5    1
    0   42 1043 4982 7890 5019 1469  248   35    5    1
    ...
Row n = 4 counts the following representatives:
  .  {{1,1,1,1}}  {{1},{1,1,1}}  {{1},{2},{1,1}}  {{1},{2},{3},{4}}
     {{1,1,1,2}}  {{1},{1,1,2}}  {{1},{2},{1,2}}
     {{1,1,2,2}}  {{1},{1,2,2}}  {{1},{2},{1,3}}
     {{1,1,2,3}}  {{1},{1,2,3}}  {{1},{2},{3,3}}
     {{1,2,3,4}}  {{1},{2,2,2}}  {{1},{2},{3,4}}
                  {{1},{2,2,3}}
                  {{1},{2,3,4}}
                  {{1,1},{1,2}}
                  {{1,1},{2,2}}
                  {{1,1},{2,3}}
                  {{1,2},{1,3}}
                  {{1,2},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A316980, connected case A319557.
For multiset partitions we have A317533, connected A322133.
Counting connected components instead of edges gives A321194.
For normal multiset partitions we have A330787, row sums A317776.
For set multipartitions we have A334550.
For set-systems we have A368096, row-sums A283877 (connected A300913).
A000110 counts set-partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A049311 counts non-isomorphic set multipartitions, connected A056156.
A058891 counts set-systems, unlabeled A000612, connected A323818.
Cf. A255903, A296122, A302545, A306005, A317532, A317775, A317794, A317795, A319560, A368094, A368095.

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@@#&&Length[#]==k&]]], {n,0,5},{k,0,n}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
G(n)={my(s=0); forpart(q=n, my(p=sum(t=1, n, y^t*subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*exp(p-subst(subst(p, x, x^2), y, y^2))); s/n!}
T(n)={[Vecrev(p) | p <- Vec(G(n))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

A368726 Number of non-isomorphic connected multiset partitions of weight n into singletons or pairs.

Original entry on oeis.org

1, 1, 3, 3, 8, 10, 26, 38, 93, 161, 381, 732, 1721, 3566, 8369, 18316, 43280, 98401, 234959, 549628, 1327726, 3175670, 7763500, 18905703, 46762513, 115613599, 289185492, 724438500, 1831398264, 4641907993, 11853385002, 30365353560

Offset: 0

Gus Wiseman, Jan 06 2024

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 10 multiset partitions:
  {{1}}  {{1,1}}    {{1},{1,1}}    {{1,1},{1,1}}      {{1},{1,1},{1,1}}
         {{1,2}}    {{2},{1,2}}    {{1,2},{1,2}}      {{1},{1,2},{2,2}}
         {{1},{1}}  {{1},{1},{1}}  {{1,2},{2,2}}      {{2},{1,2},{1,2}}
                                   {{1,3},{2,3}}      {{2},{1,2},{2,2}}
                                   {{1},{1},{1,1}}    {{2},{1,3},{2,3}}
                                   {{1},{2},{1,2}}    {{3},{1,3},{2,3}}
                                   {{2},{2},{1,2}}    {{1},{1},{1},{1,1}}
                                   {{1},{1},{1},{1}}  {{1},{2},{2},{1,2}}
                                                      {{2},{2},{2},{1,2}}
                                                      {{1},{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

For edges of any size we have A007718.
This is the connected case of A320663.
The case of singletons and strict pairs is A368727, Euler transform A339888.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A007716 counts non-isomorphic multiset partitions, into pairs A007717.
A062740 counts connected loop-graphs, unlabeled A054921.
A320732 counts factorizations into primes or semiprimes, strict A339839.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A001515, A000666, A122848, A283877, A302545, A320462, A321405, A368186, A368598, A368599, A368731.

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]}];
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]]]]]]]]];
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], Max@@Length/@#<=2&&Length[csm[#]]<=1&]]],{n,0,8}]

Inverse Euler transform of A320663.

A368727 Number of non-isomorphic connected multiset partitions of weight n into singletons or strict pairs.

Original entry on oeis.org

1, 1, 2, 2, 5, 6, 15, 21, 49, 82, 184, 341, 766, 1530, 3428, 7249, 16394, 36009, 82492, 186485, 433096, 1001495, 2358182, 5554644, 13255532, 31718030, 76656602, 185982207, 454889643, 1117496012, 2764222322, 6868902152, 17172601190

Offset: 0

Gus Wiseman, Jan 06 2024

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 15 multiset partitions:
  {1}  {12}    {2}{12}    {12}{12}      {2}{12}{12}      {12}{12}{12}
       {1}{1}  {1}{1}{1}  {13}{23}      {2}{13}{23}      {12}{13}{23}
                          {1}{2}{12}    {3}{13}{23}      {13}{23}{23}
                          {2}{2}{12}    {1}{2}{2}{12}    {13}{24}{34}
                          {1}{1}{1}{1}  {2}{2}{2}{12}    {14}{24}{34}
                                        {1}{1}{1}{1}{1}  {1}{2}{12}{12}
                                                         {1}{2}{13}{23}
                                                         {2}{2}{12}{12}
                                                         {2}{2}{13}{23}
                                                         {2}{3}{13}{23}
                                                         {3}{3}{13}{23}
                                                         {1}{1}{2}{2}{12}
                                                         {1}{2}{2}{2}{12}
                                                         {2}{2}{2}{2}{12}
                                                         {1}{1}{1}{1}{1}{1}

Andrew Howroyd, Table of n, a(n) for n = 0..50

For edges of any size we have A056156, with loops A007718.
This is the connected case of A339888.
Allowing loops {x,x} gives A368726, Euler transform A320663.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A007716 counts non-isomorphic multiset partitions, into pairs A007717.
A062740 counts connected loop-graphs, unlabeled A054921.
A320732 counts factorizations into primes or semiprimes, strict A339839.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A001515, A000666, A122848, A283877, A302545, A320462, A321405, A368598, A368599, A368731.

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]}];
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]]]]]]]]];
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@@@#&&Max@@Length/@#<=2&&Length[csm[#]]<=1&]]],{n,0,8}]

Inverse Euler transform of A339888.

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.

A320291 Number of singleton-free multiset partitions of integer partitions of n with no 1's.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 3, 3, 7, 8, 15, 19, 36, 46, 79, 110, 181, 254, 407, 580, 907, 1309, 2004, 2909, 4410, 6407, 9599, 13984, 20782, 30252, 44677, 64967, 95414, 138563, 202527, 293583, 427442, 618337, 897023, 1295020, 1872696, 2697777, 3889964, 5591917, 8041593, 11535890

Offset: 0

Gus Wiseman, Oct 09 2018

nonn

			The a(4) = 1 through a(10) = 15 multiset partitions:
  ((22))  ((23))  ((24))   ((25))   ((26))      ((27))      ((28))
                  ((33))   ((34))   ((35))      ((36))      ((37))
                  ((222))  ((223))  ((44))      ((45))      ((46))
                                    ((224))     ((225))     ((55))
                                    ((233))     ((234))     ((226))
                                    ((2222))    ((333))     ((235))
                                    ((22)(22))  ((2223))    ((244))
                                                ((22)(23))  ((334))
                                                            ((2224))
                                                            ((2233))
                                                            ((22222))
                                                            ((22)(24))
                                                            ((22)(33))
                                                            ((23)(23))
                                                            ((22)(222))

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A002865, A007716, A049311, A083751, A283877, A293606, A302545, A304966, A304967, A320294, A320295, A320296.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@Select[IntegerPartitions[n],FreeQ[#,1]&],FreeQ[Length/@#,1]&]],{n,20}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=vector(n,i,i>1)); concat([1], EulerT(EulerT(v)-v))} \\ Andrew Howroyd, Oct 25 2018

Euler transform of A083751. - Andrew Howroyd, Oct 25 2018

Terms a(21) and beyond from Andrew Howroyd, Oct 25 2018

A321412 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons and with aperiodic parts.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 3, 4, 12, 20, 42

Offset: 0

Gus Wiseman, Nov 16 2018

A multiset is aperiodic if its multiplicities are relatively prime.
Also the number of nonnegative integer symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with no row or column having a common divisor > 1 or summing to 1.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
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(5) = 1 through a(8) = 12 multiset partitions:
{{12}{12}}  {{12}{122}}  {{112}{122}}    {{112}{1222}}    {{1112}{1222}}
                         {{12}{1222}}    {{12}{12222}}    {{112}{12222}}
                         {{12}{13}{23}}  {{12}{13}{233}}  {{12}{122222}}
                                         {{13}{23}{123}}  {{122}{11222}}
                                                          {{12}{123}{233}}
                                                          {{12}{13}{2333}}
                                                          {{13}{112}{233}}
                                                          {{13}{122}{233}}
                                                          {{13}{23}{1233}}
                                                          {{23}{123}{123}}
                                                          {{12}{12}{34}{34}}
                                                          {{12}{13}{24}{34}}

Cf. A000219, A007716, A120733, A138178, A302545, A316983, A319616.

Cf. A320796, A320797, A320803, A320806, A320809, A320813, A321408, A321410, A321411.

cp's OEIS Frontend

A321413 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons and relatively prime part sizes.

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A321677 Number of non-isomorphic set multipartitions (multisets of sets) of weight n with no singletons.

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A330124 Number of unlabeled set-systems with n vertices and no endpoints.

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A368096 Triangle read by rows where T(n,k) is the number of non-isomorphic set-systems of length k and weight n.

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A368099 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element sets of finite nonempty multisets with cardinalities summing to n, or strict multiset partitions of weight n and length k.

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A368726 Number of non-isomorphic connected multiset partitions of weight n into singletons or pairs.

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A368727 Number of non-isomorphic connected multiset partitions of weight n into singletons or strict pairs.

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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.

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A320291 Number of singleton-free multiset partitions of integer partitions of n with no 1's.

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