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

0,8

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 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.

Links

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

Example

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}}

Crossrefs

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

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

Sequence in context: A025618 A133572 A121852 * A242251 A122537 A059736

Adjacent sequences: A321408 A321409 A321410 * A321412 A321413 A321414

Keyword

nonn,more

Author

Gus Wiseman, Nov 16 2018

Status

approved