A323581 Number of ways to fill a Young diagram with positive integers summing to n such that the rows are strictly increasing and the columns are strictly decreasing.

1, 1, 1, 3, 3, 5, 8, 10, 14, 19, 28, 34, 48, 60, 80, 106, 134, 171, 222, 279, 354, 452, 562, 706, 884, 1100

Offset: 0

Author

Gus Wiseman, Jan 18 2019

Keywords

• nonn
• more

Examples

			The a(8) = 14 tableaux:
  8   1 7   2 6   3 5   1 2 5   1 3 4
.
  7   6   5   2 5   3 4   2 3
  1   2   3   1     1     1 2
.
  5   4
  2   3
  1   1
		

Links

• nLab, Young Diagram.

Crossrefs

Cf. A000085, A000219, A003293, A114736, A138178, A299968, A323436, A323437, A323438, A323439, A323580.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Sum[Length[Select[Reverse/@Sort/@Map[primeMS,sqfacs[y],{2}],And@@Greater@@@DeleteCases[Transpose[PadRight[#]],0,{2}]&]],{y,Times@@Prime/@#&/@IntegerPartitions[n]}],{n,10}]