OFFSET
0,4
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.
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.
EXAMPLE
Non-isomorphic representatives of the a(1) = 1 through a(6) = 15 multiset partitions:
{1} {1}{2} {2}{12} {2}{122} {12}{122} {2}{12222}
{1}{2}{3} {1}{1}{23} {2}{1222} {1}{23}{233}
{1}{3}{23} {1}{23}{23} {1}{3}{2333}
{1}{2}{3}{4} {1}{3}{233} {2}{13}{233}
{2}{13}{23} {3}{23}{123}
{3}{3}{123} {3}{3}{1233}
{1}{2}{2}{34} {1}{1}{1}{234}
{1}{2}{4}{34} {1}{2}{34}{34}
{1}{2}{3}{4}{5} {1}{2}{4}{344}
{1}{3}{24}{34}
{1}{4}{4}{234}
{2}{4}{12}{34}
{1}{2}{3}{3}{45}
{1}{2}{3}{5}{45}
{1}{2}{3}{4}{5}{6}
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 16 2018
STATUS
approved