A323586 - OEIS

A323586

Number of plane partitions of n with no repeated rows (or, equivalently, no repeated columns).

%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 .

%e 2 11

%e 1 1

%e .

%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