A323439 Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are strictly increasing.

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 0, 1, 2, 2, 0, 1, 1, 1, 0, 2, 2, 1, 0, 0, 2, 0, 0, 1, 4, 1, 0, 2, 2, 2, 0, 1, 2, 2, 0, 1, 4, 1, 0, 0, 2, 1, 0, 0, 1, 2, 0, 1, 0, 2, 0, 2, 2, 1, 0, 1, 2, 0, 0, 2, 4, 1, 0, 2, 4, 1, 0, 1, 2, 1, 0, 2, 4, 1, 0, 0, 2, 1, 0, 2, 2, 2

Offset

1,6

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..87.

Wikipedia, Young tableau

Formula

Sum_{A056239(n) = k} a(k) = A323451(n).

Example

The a(630) = 8 tableaux:
  123   124   1234
  24    23    2
.
  12   12   123   124
  23   24   2     2
  4    3    4     3
.
  12
  2
  3
  4

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@@(Less@@@DeleteCases[Transpose[PadRight[#]], 0, {2}]), And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])]&]], {y, 100}]

Crossrefs

Cf. A000085, A000219, A056239, A112798, A114736, A117433, A299968, A323300, A323430, A323436, A323438, A323450, A323451.

Sequence in context: A344478 A335489 A281188 * A054008 A347031 A125676

Adjacent sequences: A323436 A323437 A323438 * A323440 A323441 A323442

Keyword

nonn

Author

Gus Wiseman, Jan 16 2019

Status

approved