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%I #13 Jun 21 2021 23:30:37
%S 1,1,1,4,4,9,17,28,41,75,122,192,314,484,771,1216,1861,2848,4395,6610,
%T 10037
%N Number of non-isomorphic weight-n chains of distinct multisets whose dual is also a chain of distinct multisets.
%C The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
%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.
%C From _Gus Wiseman_, Jan 17 2019: (Start)
%C Also the number of plane partitions of n with no repeated rows or columns. For example, the a(6) = 17 plane partitions are:
%C 6 51 42 321
%C .
%C 5 4 41 31 32 31 22 221 211
%C 1 2 1 2 1 11 2 1 11
%C .
%C 3 21 21 111
%C 2 2 11 11
%C 1 1 1 1
%C (End)
%e Non-isomorphic representatives of the a(1) = 1 through a(5) = 9 chains:
%e 1: {{1}}
%e 2: {{1,1}}
%e 3: {{1,1,1}}
%e {{1,2,2}}
%e {{1},{1,1}}
%e {{2},{1,2}}
%e 4: {{1,1,1,1}}
%e {{1,2,2,2}}
%e {{1},{1,1,1}}
%e {{2},{1,2,2}}
%e 5: {{1,1,1,1,1}}
%e {{1,1,2,2,2}}
%e {{1,2,2,2,2}}
%e {{1},{1,1,1,1}}
%e {{2},{1,1,2,2}}
%e {{2},{1,2,2,2}}
%e {{1,1},{1,1,1}}
%e {{1,2},{1,2,2}}
%e {{2,2},{1,2,2}}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
%t Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[UnsameQ@@#,UnsameQ@@Transpose[PadRight[#]],And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,IntegerPartitions[n]}],{n,10}] (* _Gus Wiseman_, Jan 18 2019 *)
%Y Cf. A000219, A003293, A007716, A059201, A283877, A316980, A316983, A318099, A319558, A319616-A319646.
%Y Cf. A000085, A138178, A323436.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Sep 25 2018
%E a(11)-a(17) from _Gus Wiseman_, Jan 18 2019
%E a(18)-a(21) from _Robert Price_, Jun 21 2021
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