%I #5 Jan 20 2019 23:19:38
%S 1,1,2,5,8,16,30,53,89,158,265,443,735,1197
%N Number of plane partitions of n with no repeated rows (or, equivalently, no repeated columns).
%e The a(4) = 8 plane partitions with no repeated rows:
%e 4 31 22 211 1111
%e .
%e 3 21 111
%e 1 1 1
%e The a(6) = 30 plane partitions with no repeated columns:
%e 6 51 42 321
%e .
%e 5 4 41 3 31 32 31 22 21 221 211
%e 1 2 1 3 2 1 11 2 21 1 11
%e .
%e 4 3 31 2 21 22 21 111
%e 1 2 1 2 2 1 11 11
%e 1 1 1 2 1 1 1 1
%e .
%e 3 2 21 11
%e 1 2 1 11
%e 1 1 1 1
%e 1 1 1 1
%e .
%e 2 11
%e 1 1
%e 1 1
%e 1 1
%e 1 1
%e .
%e 1
%e 1
%e 1
%e 1
%e 1
%e 1
%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@@#,And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,IntegerPartitions[n]}],{n,10}]
%Y Cf. A000219, A003293 (strict rows), A114736 (strict rows and columns), A117433 (distinct entries), A299968, A319646 (no repeated rows or columns), A323429, A323436 (plane partitions of type), A323580, A323587.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Jan 20 2019