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

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

Offset

0,4

Comments

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.

Links

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

Example

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}}

Crossrefs

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

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

Sequence in context: A130596 A280730 A339848 * A151954 A032247 A052281

Adjacent sequences: A321406 A321407 A321408 * A321410 A321411 A321412

Keyword

nonn,more

Author

Gus Wiseman, Nov 16 2018

Status

approved