A320804 -id:A320804 - cp's OEIS frontend

A320797 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons.

1, 0, 1, 1, 3, 4, 9, 15, 33, 60, 121

Offset: 0

Gus Wiseman, Nov 02 2018

Also the number of nonnegative integer square symmetric matrices with sum of elements equal to n and no rows or columns summing to 0 or 1, up to row and column permutations.

			Non-isomorphic representatives of the a(2) = 1 through a(7) = 15 multiset partitions:
  {{11}}  {{111}}  {{1111}}    {{11111}}    {{111111}}      {{1111111}}
                   {{11}{22}}  {{11}{122}}  {{111}{222}}    {{111}{1222}}
                   {{12}{12}}  {{11}{222}}  {{112}{122}}    {{111}{2222}}
                               {{12}{122}}  {{11}{2222}}    {{112}{1222}}
                                            {{12}{1222}}    {{11}{22222}}
                                            {{22}{1122}}    {{12}{12222}}
                                            {{11}{22}{33}}  {{122}{1122}}
                                            {{11}{23}{23}}  {{22}{11222}}
                                            {{12}{13}{23}}  {{11}{12}{233}}
                                                            {{11}{22}{233}}
                                                            {{11}{22}{333}}
                                                            {{11}{23}{233}}
                                                            {{12}{12}{333}}
                                                            {{12}{13}{233}}
                                                            {{13}{23}{123}}
Inequivalent representatives of the a(6) = 9 symmetric matrices with no rows or columns summing to 1:
  [6]
.
  [3 0]  [2 1]  [4 0]  [3 1]  [2 2]
  [0 3]  [1 2]  [0 2]  [1 1]  [2 0]
.
  [2 0 0]  [2 0 0]  [1 1 0]
  [0 2 0]  [0 1 1]  [1 0 1]
  [0 0 2]  [0 1 1]  [0 1 1]

Cf. A000219, A007716, A302545, A316980, A316983, A319560, A319616, A319721.

Cf. A320796, A320798, A320799, A320804, A320811, A320812, A320813.

A320813 Number of non-isomorphic multiset partitions of an aperiodic multiset of weight n such that there are no singletons and all parts are themselves aperiodic multisets.

Original entry on oeis.org

1, 0, 1, 2, 5, 13, 33, 104, 293, 938, 2892

Offset: 0

Gus Wiseman, Nov 08 2018

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which (1) the row sums are all > 1, (2) the positive entries in each row are relatively prime, and (3) the column-sums are relatively prime.
A multiset is aperiodic if its multiplicities are relatively prime.
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(5) = 13 multiset partitions:
  {{1,2}}  {{1,2,2}}  {{1,2,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,3,3}}    {{1,2,2,2,2}}
                      {{1,2,3,4}}    {{1,2,2,3,3}}
                      {{1,2},{3,4}}  {{1,2,3,3,3}}
                      {{1,3},{2,3}}  {{1,2,3,4,4}}
                                     {{1,2,3,4,5}}
                                     {{1,2},{1,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,2},{3,4,4}}
                                     {{1,2},{3,4,5}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,3},{1,2,3}}

This is the case of A320804 where the underlying multiset is aperiodic.

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A302545, A303386, A303546, A303707, A303708, A320797-A320813, A321283, A321390.

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]}]]];
aperQ[m_]:=Length[m]==0||GCD@@Length/@Split[Sort[m]]==1;
Table[Length[Union[brute /@ Select[mpm[n],And[Min@@Length/@#>1,aperQ[Join@@#]&&And@@aperQ /@ #]&]]],{n,0,7}] (* Gus Wiseman, Jan 19 2024 *)

Definition corrected by Gus Wiseman, Jan 19 2024

A320799 Number of non-isomorphic (not necessarily strict) antichains of multisets of weight n with no singletons or leaves (vertices that appear only once).

Original entry on oeis.org

1, 0, 1, 1, 5, 4, 22, 27, 107, 212, 689

Offset: 0

Gus Wiseman, Nov 02 2018

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(7) = 27 multiset partitions:
  {{11}}  {{111}}  {{1111}}    {{11111}}    {{111111}}      {{1111111}}
                   {{1122}}    {{11222}}    {{111222}}      {{1112222}}
                   {{11}{11}}  {{11}{122}}  {{112222}}      {{1122222}}
                   {{11}{22}}  {{11}{222}}  {{112233}}      {{1122333}}
                   {{12}{12}}               {{111}{111}}    {{111}{1222}}
                                            {{11}{1222}}    {{11}{12222}}
                                            {{111}{222}}    {{111}{2222}}
                                            {{112}{122}}    {{11}{12233}}
                                            {{11}{2222}}    {{111}{2233}}
                                            {{112}{222}}    {{112}{1222}}
                                            {{11}{2233}}    {{11}{22222}}
                                            {{112}{233}}    {{112}{2222}}
                                            {{122}{122}}    {{11}{22333}}
                                            {{123}{123}}    {{112}{2333}}
                                            {{11}{11}{11}}  {{113}{2233}}
                                            {{11}{12}{22}}  {{122}{1233}}
                                            {{11}{22}{22}}  {{222}{1122}}
                                            {{11}{22}{33}}  {{11}{11}{122}}
                                            {{11}{23}{23}}  {{11}{11}{222}}
                                            {{12}{12}{12}}  {{11}{12}{222}}
                                            {{12}{12}{22}}  {{11}{12}{233}}
                                            {{12}{13}{23}}  {{11}{22}{233}}
                                                            {{11}{22}{333}}
                                                            {{12}{12}{222}}
                                                            {{12}{12}{233}}
                                                            {{12}{12}{333}}
                                                            {{12}{13}{233}}

Cf. A006126, A096827, A285572, A293993, A293994, A302545, A318099, A319560, A319719, A319721.

Cf. A320797, A320798, A320804, A320811, A320812, A320813.

A321408 Number of non-isomorphic self-dual multiset partitions of weight n whose parts are aperiodic.

Original entry on oeis.org

1, 1, 1, 2, 5, 9, 18, 35, 75, 153, 318

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, in which no row or column has a common divisor > 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(1) = 1 through a(6) = 18 multiset partitions:
  {1}  {1}{2}  {2}{12}    {12}{12}      {12}{122}        {112}{122}
               {1}{2}{3}  {2}{122}      {2}{1222}        {12}{1222}
                          {1}{1}{23}    {1}{23}{23}      {2}{12222}
                          {1}{3}{23}    {1}{3}{233}      {12}{13}{23}
                          {1}{2}{3}{4}  {2}{13}{23}      {1}{23}{233}
                                        {3}{3}{123}      {1}{3}{2333}
                                        {1}{2}{2}{34}    {2}{13}{233}
                                        {1}{2}{4}{34}    {3}{23}{123}
                                        {1}{2}{3}{4}{5}  {3}{3}{1233}
                                                         {1}{1}{1}{234}
                                                         {1}{2}{34}{34}
                                                         {1}{2}{4}{344}
                                                         {1}{3}{24}{34}
                                                         {1}{4}{4}{234}
                                                         {2}{4}{12}{34}
                                                         {1}{2}{3}{3}{45}
                                                         {1}{2}{3}{5}{45}
                                                         {1}{2}{3}{4}{5}{6}

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

Cf. A320796, A320797, A320803, A320804, A320805, A320806, A320807, A320809, A320813, A321410, A321411, A321412.

A321402 Number of non-isomorphic strict self-dual multiset partitions of weight n with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 8, 14, 27, 53, 105

Offset: 0

Gus Wiseman, Nov 09 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, in which the rows are all different and none sums 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(2) = 1 through a(7) = 14 multiset partitions:
  {{11}}  {{111}}  {{1111}}    {{11111}}    {{111111}}      {{1111111}}
                   {{11}{22}}  {{11}{122}}  {{111}{222}}    {{111}{1222}}
                               {{11}{222}}  {{112}{122}}    {{111}{2222}}
                               {{12}{122}}  {{11}{2222}}    {{112}{1222}}
                                            {{12}{1222}}    {{11}{22222}}
                                            {{22}{1122}}    {{12}{12222}}
                                            {{11}{22}{33}}  {{122}{1122}}
                                            {{12}{13}{23}}  {{22}{11222}}
                                                            {{11}{12}{233}}
                                                            {{11}{22}{233}}
                                                            {{11}{22}{333}}
                                                            {{11}{23}{233}}
                                                            {{12}{13}{233}}
                                                            {{13}{23}{123}}

Cf. A000219, A007716, A059201, A302545, A316980, A316983, A319560, A319616.

Cf. A320796, A320797, A320798, A320804, A320811, A320812, A321401, A321404, A321405, A321406, A321407.

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

cp's OEIS Frontend

A320797 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons.

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A320813 Number of non-isomorphic multiset partitions of an aperiodic multiset of weight n such that there are no singletons and all parts are themselves aperiodic multisets.

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A320799 Number of non-isomorphic (not necessarily strict) antichains of multisets of weight n with no singletons or leaves (vertices that appear only once).

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A321408 Number of non-isomorphic self-dual multiset partitions of weight n whose parts are aperiodic.

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A321402 Number of non-isomorphic strict self-dual multiset partitions of weight n with no singletons.

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

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