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A323581
Number of ways to fill a Young diagram with positive integers summing to n such that the rows are strictly increasing and the columns are strictly decreasing.
1
1, 1, 1, 3, 3, 5, 8, 10, 14, 19, 28, 34, 48, 60, 80, 106, 134, 171, 222, 279, 354, 452, 562, 706, 884, 1100
OFFSET
0,4
EXAMPLE
The a(8) = 14 tableaux:
8 1 7 2 6 3 5 1 2 5 1 3 4
.
7 6 5 2 5 3 4 2 3
1 2 3 1 1 1 2
.
5 4
2 3
1 1
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
sqfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[sqfacs[n/d], Min@@#>=d&]], {d, Select[Rest[Divisors[n]], SquareFreeQ]}]];
Table[Sum[Length[Select[Reverse/@Sort/@Map[primeMS, sqfacs[y], {2}], And@@Greater@@@DeleteCases[Transpose[PadRight[#]], 0, {2}]&]], {y, Times@@Prime/@#&/@IntegerPartitions[n]}], {n, 10}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jan 18 2019
STATUS
approved