|
%I #4 Nov 07 2018 21:45:53
%S 1,1,3,6,17,41,122,345,1077,3385,11214
%N Number of non-isomorphic multiset partitions of weight n in which all parts are aperiodic and all parts of the dual are also aperiodic.
%C Also the number of nonnegative integer matrices up to row and column permutations with sum of entries equal to n and no zero rows or columns, in which each row and each column has relatively prime nonzero entries.
%C 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}}.
%C A multiset is aperiodic if its multiplicities are relatively prime.
%C The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
%e Non-isomorphic representatives of the a(1) = 1 through a(4) = 17 multiset partitions:
%e {{1}} {{1,2}} {{1,2,3}} {{1,2,3,4}}
%e {{1},{1}} {{1},{2,3}} {{1,2},{1,2}}
%e {{1},{2}} {{2},{1,2}} {{1},{2,3,4}}
%e {{1},{1},{1}} {{1,2},{3,4}}
%e {{1},{2},{2}} {{1,3},{2,3}}
%e {{1},{2},{3}} {{2},{1,2,2}}
%e {{3},{1,2,3}}
%e {{1},{1},{2,3}}
%e {{1},{2},{1,2}}
%e {{1},{2},{3,4}}
%e {{1},{3},{2,3}}
%e {{2},{2},{1,2}}
%e {{1},{1},{1},{1}}
%e {{1},{1},{2},{2}}
%e {{1},{2},{2},{2}}
%e {{1},{2},{3},{3}}
%e {{1},{2},{3},{4}}
%Y Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303546, A303707, A303708, A316983, A320800-A320810.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Nov 07 2018
|
