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A323439
Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are strictly increasing.
7
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.
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 16 2019
STATUS
approved