A323580 Number of ways to fill a Young diagram with positive integers summing to n such that the rows are weakly decreasing and the columns are weakly increasing.

1, 1, 3, 6, 13, 23, 45, 76, 136, 225, 381, 611, 1001, 1570, 2489, 3842, 5948, 9022, 13714, 20501, 30649, 45262, 66721, 97393, 141888, 204993

Offset

0,3

Links

Table of n, a(n) for n=0..25.

nLab, Young Diagram.

Example

The a(5) = 23 tableaux:
  5   41   32   311   221   2111   11111
.
  1   2   11   21   11   111   111   1111
  4   3   3    2    21   2     11    1
.
  1   1   11   11   111
  1   2   1    11   1
  3   2   2    1    1
.
  1   11
  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]]}]];
Table[Sum[Length[Select[Reverse/@Sort/@Map[primeMS, facs[y], {2}], And@@(GreaterEqual@@@Transpose[PadRight[#]])&]], {y, Times@@Prime/@#&/@IntegerPartitions[n]}], {n, 10}]

Crossrefs

Cf. A000085, A000219, A003293, A114736, A138178, A299968, A323436, A323437, A323438, A323439, A323581.

Sequence in context: A308747 A022811 A295730 * A002799 A285263 A162426

Adjacent sequences: A323577 A323578 A323579 * A323581 A323582 A323583

Keyword

nonn,more

Author

Gus Wiseman, Jan 18 2019

Status

approved