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%I #9 Sep 24 2018 08:57:47
%S 1,1,3,7,21,55,169,496,1582,5080,17073
%N The squarefree dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted without multiplicity. Then a(n) is the number of non-isomorphic multiset partitions of weight n whose squarefree dual is strict (no repeated blocks).
%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, a(2) = 3, and a(3) = 7 multiset partitions:
%e 1: {{1}}
%e 2: {{1,1}}
%e {{1},{1}}
%e {{1},{2}}
%e 3: {{1,1,1}}
%e {{1},{1,1}}
%e {{1},{2,2}}
%e {{2},{1,2}}
%e {{1},{1},{1}}
%e {{1},{2},{2}}
%e {{1},{2},{3}}
%Y Cf. A007716, A007718, A049311, A053419, A056156, A059201, A283877, A316980.
%Y Cf. A319557, A319559, A319560, A319564, A319565, A319566, A319567.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Sep 23 2018
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