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

0,3

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.

Links

Table of n, a(n) for n=0..26.

nLab, Young Diagram.

The Unapologetic Mathematician weblog, Generalized Young Tableaux.

Example

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

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}]

Crossrefs

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

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

Sequence in context: A049940 A265947 A358831 * A051749 A278788 A279828

Adjacent sequences: A323447 A323448 A323449 * A323451 A323452 A323453

Keyword

nonn,more

Author

Gus Wiseman, Jan 16 2019

Extensions

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

Status

approved