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

0,4

Links

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

nLab, Young Diagram.

Example

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

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}]

Crossrefs

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

Sequence in context: A342343 A116645 A177739 * A327731 A193744 A039872

Adjacent sequences: A323578 A323579 A323580 * A323582 A323583 A323584

Keyword

nonn,more

Author

Gus Wiseman, Jan 18 2019

Status

approved