A323437 Number of semistandard Young tableaux whose entries are the prime indices of n.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 5, 1, 2, 2, 1, 2, 4, 1, 2, 2, 4, 1, 3, 1, 2, 2, 2, 2, 4, 1, 2, 1, 2, 1, 5, 2, 2, 2

Offset

0,7

Comments

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

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.

Is this a duplicate of A339887? - R. J. Mathar, Feb 03 2021

Links

Table of n, a(n) for n=0..87.

Formula

Sum_{A056239(n) = k} a(k) = A003293(n).

Example

The a(60) = 5 tableaux:
  1123
.
  11   112   113
  23   3     2
.
  11
  2
  3

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

Crossrefs

Cf. A000085, A000219, A003293, A053529, A056239, A112798, A138178, A153452, A296188.

Cf. A323300, A323429, A323432, A323436, A323438, A323439.

Sequence in context: A096825 A318369 A007875 * A339887 A259936 A354870

Adjacent sequences: A323434 A323435 A323436 * A323438 A323439 A323440

Keyword

nonn

Author

Gus Wiseman, Jan 15 2019

Status

approved