A323450 Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are weakly increasing.

1, 1, 3, 6, 14, 26, 56, 103, 203, 374, 702, 1262, 2306, 4078, 7242, 12628, 21988, 37756, 64682, 109606, 185082, 309958, 516932, 856221, 1412461, 2316416, 3783552

Offset: 0

Author

Gus Wiseman, Jan 16 2019

Keywords

• nonn
• more

Comments

A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers.

Examples

			The a(4) = 14 generalized Young tableaux:
  4   1 3   2 2   1 1 2   1 1 1 1
.
  1   2   1 1   1 2   1 1   1 1 1
  3   2   2     1     1 1   1
.
  1   1 1
  1   1
  2   1
.
  1
  1
  1
  1
The a(5) = 26 generalized Young tableaux:
  5   1 4   2 3   1 1 3   1 2 2   1 1 1 2   1 1 1 1 1
.
  1   2   1 1   1 3   1 2   1 1   1 1 1   1 1 2   1 1 1   1 1 1 1
  4   3   3     1     2     1 2   2       1       1 1     1
.
  1   1   1 1   1 2   1 1   1 1 1
  1   2   1     1     1 1   1
  3   2   2     1     1     1
.
  1   1 1
  1   1
  1   1
  2   1
.
  1
  1
  1
  1
  1
		

Links

• nLab, Young Diagram.
• The Unapologetic Mathematician weblog, Generalized Young Tableaux.

Crossrefs

Cf. A000085, A000219, A003293, A053529, A114736, A138178, A296188, A299968.

Cf. A323436, A323437, A323438, A323439, A323451.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])&]],{y,IntegerPartitions[n]}],{n,10}]

Extensions

a(16)-a(26) from Seiichi Manyama, Aug 19 2020