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A323438
Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are weakly increasing.
13
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.
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 16 2019
STATUS
approved