A323438 - OEIS

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

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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

2 1

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