A321408 -id:A321408 - cp's OEIS frontend

A321411 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons, with aperiodic parts whose sizes are relatively prime.

1, 0, 0, 0, 0, 1, 0, 4, 6, 16, 25

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 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(9) = 16 multiset partitions:
  {{12}{122}}  {{112}{1222}}    {{112}{12222}}    {{1112}{11222}}
               {{12}{12222}}    {{122}{11222}}    {{1112}{12222}}
               {{12}{13}{233}}  {{12}{123}{233}}  {{12}{1222222}}
               {{13}{23}{123}}  {{13}{112}{233}}  {{12}{123}{2333}}
                                {{13}{122}{233}}  {{12}{13}{23333}}
                                {{23}{123}{123}}  {{12}{223}{1233}}
                                                  {{13}{112}{2333}}
                                                  {{13}{223}{1233}}
                                                  {{13}{23}{12333}}
                                                  {{23}{122}{1233}}
                                                  {{23}{123}{1233}}
                                                  {{12}{12}{34}{234}}
                                                  {{12}{12}{34}{344}}
                                                  {{12}{13}{14}{234}}
                                                  {{12}{13}{24}{344}}
                                                  {{12}{14}{34}{234}}

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

Cf. A320796, A320797, A320803, A320806, A320809, A320813, A321283, A321408-A321413.

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

Original entry on oeis.org

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.

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

A321411 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons, with aperiodic parts whose sizes are relatively prime.

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

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