A320805 -id:A320805 - cp's OEIS frontend

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

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.

A321409 Number of non-isomorphic self-dual multiset partitions of weight n whose part sizes are relatively prime.

Original entry on oeis.org

1, 1, 1, 3, 6, 16, 27, 71, 135, 309, 621

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).
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(5) = 16 multiset partitions:
  {{1}}  {{1}{2}}  {{1}{22}}    {{1}{222}}      {{11}{122}}
                   {{2}{12}}    {{2}{122}}      {{11}{222}}
                   {{1}{2}{3}}  {{1}{1}{23}}    {{12}{122}}
                                {{1}{2}{33}}    {{1}{2222}}
                                {{1}{3}{23}}    {{2}{1222}}
                                {{1}{2}{3}{4}}  {{1}{22}{33}}
                                                {{1}{23}{23}}
                                                {{1}{2}{333}}
                                                {{1}{3}{233}}
                                                {{2}{12}{33}}
                                                {{2}{13}{23}}
                                                {{3}{3}{123}}
                                                {{1}{2}{2}{34}}
                                                {{1}{2}{3}{44}}
                                                {{1}{2}{4}{34}}
                                                {{1}{2}{3}{4}{5}}

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

Cf. A320796, A320797, A320800, A320805, A320806, A320809, A320811, A320813, A321283, A321410, A321411, A321413.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

A321409 Number of non-isomorphic self-dual multiset partitions of weight n whose part sizes are relatively prime.

Views

Author

Keywords

Comments

Examples

Crossrefs