A321413 -id:A321413 - 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.

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

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

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