A321412 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons and with aperiodic parts.

1, 0, 0, 0, 1, 1, 3, 4, 12, 20, 42

Offset

0,7

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 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(8) = 12 multiset partitions:
{{12}{12}}  {{12}{122}}  {{112}{122}}    {{112}{1222}}    {{1112}{1222}}
                         {{12}{1222}}    {{12}{12222}}    {{112}{12222}}
                         {{12}{13}{23}}  {{12}{13}{233}}  {{12}{122222}}
                                         {{13}{23}{123}}  {{122}{11222}}
                                                          {{12}{123}{233}}
                                                          {{12}{13}{2333}}
                                                          {{13}{112}{233}}
                                                          {{13}{122}{233}}
                                                          {{13}{23}{1233}}
                                                          {{23}{123}{123}}
                                                          {{12}{12}{34}{34}}
                                                          {{12}{13}{24}{34}}

Crossrefs

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

Cf. A320796, A320797, A320803, A320806, A320809, A320813, A321408, A321410, A321411.

Sequence in context: A124817 A306005 A124818 * A124638 A124639 A279696

Adjacent sequences: A321409 A321410 A321411 * A321413 A321414 A321415

Keyword

nonn,more

Author

Gus Wiseman, Nov 16 2018

Status

approved