A321409 -id:A321409 - cp's OEIS frontend

A321410 Number of non-isomorphic self-dual multiset partitions of weight n whose parts are aperiodic multisets whose sizes are relatively prime.

1, 1, 1, 2, 4, 9, 15, 35, 69, 149, 301

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 relatively prime row sums (or column sums) and no row or column having 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) = 15 multiset partitions:
  {1}  {1}{2}  {2}{12}    {2}{122}      {12}{122}        {2}{12222}
               {1}{2}{3}  {1}{1}{23}    {2}{1222}        {1}{23}{233}
                          {1}{3}{23}    {1}{23}{23}      {1}{3}{2333}
                          {1}{2}{3}{4}  {1}{3}{233}      {2}{13}{233}
                                        {2}{13}{23}      {3}{23}{123}
                                        {3}{3}{123}      {3}{3}{1233}
                                        {1}{2}{2}{34}    {1}{1}{1}{234}
                                        {1}{2}{4}{34}    {1}{2}{34}{34}
                                        {1}{2}{3}{4}{5}  {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, A320803, A320806, A320809, A320813, A321283, A321408, A321409, A321411.

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A321410 Number of non-isomorphic self-dual multiset partitions of weight n whose parts are aperiodic multisets whose sizes are relatively prime.

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

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