A323438 Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are weakly increasing.

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 3, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 4, 1, 7, 2, 2, 2, 8, 1, 2, 2, 7, 1, 4, 1, 4, 4, 2, 1, 12, 2, 3, 2, 4, 1, 5, 2, 7, 2, 2, 1, 10, 1, 2, 4, 11, 2, 4, 1, 4, 2, 4, 1, 13, 1, 2, 3, 4, 2, 4, 1, 12, 5, 2, 1, 10, 2

Offset

1,4

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Links

Table of n, a(n) for n=1..85.

Wikipedia, Young tableau

Formula

Sum_{A056239(n) = k} a(k) = A323450(n).

Example

The a(96) = 19 tableaux:
  111112
.
  111   1111   1112   11111   11112
  112   12     11     2       1
.
  11   111   111   112   1111   1112
  11   11    12    11    1      1
  12   2     1     1     2      1
.
  11   11   111   112
  11   12   1     1
  1    1    1     1
  2    1    2     1
.
  11   12
  1    1
  1    1
  1    1
  2    1
.
  1
  1
  1
  1
  1
  2

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[Length[Select[ptnplane[y], And[And@@LessEqual@@@#, And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])]&]], {y, 100}]

Crossrefs

Cf. A000085, A000219, A003293, A056239, A112798, A114736, A117433, A138178, A299968, A323300, A323430, A323436, A323438, A323450.

Sequence in context: A316365 A292886 A317508 * A317141 A317791 A318559

Adjacent sequences: A323435 A323436 A323437 * A323439 A323440 A323441

Keyword

nonn

Author

Gus Wiseman, Jan 16 2019

Status

approved