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A323437 Number of semistandard Young tableaux whose entries are the prime indices of n. 7
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 (list; graph; refs; listen; history; text; internal format)
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 A050320

Adjacent sequences:  A323434 A323435 A323436 * A323438 A323439 A323440

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 15 2019

STATUS

approved

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Last modified July 31 02:33 EDT 2021. Contains 346367 sequences. (Running on oeis4.)