

A321411


Number of nonisomorphic selfdual multiset partitions of weight n with no singletons, with aperiodic parts whose sizes are relatively prime.


5



1, 0, 0, 0, 0, 1, 0, 4, 6, 16, 25
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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

Nonisomorphic 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, A321408A321413.
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



