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

0,4

Comments

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.

Links

Table of n, a(n) for n=0..10.

Example

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}

Crossrefs

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

Cf. A320796, A320803, A320806, A320809, A320813, A321283, A321408, A321409, A321411.

Sequence in context: A000879 A262232 A358830 * A218912 A230868 A014290

Adjacent sequences: A321407 A321408 A321409 * A321411 A321412 A321413

Keyword

nonn,more

Author

Gus Wiseman, Nov 16 2018

Status

approved