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A323451
Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are strictly increasing.
3
1, 1, 1, 3, 3, 6, 9, 12, 19, 27, 39, 54, 79, 107, 150, 209, 282, 387, 525, 707, 949, 1272, 1688, 2244, 2968, 3902, 5125, 6712, 8752, 11383, 14780, 19109, 24671, 31768, 40791, 52280, 66860, 85296, 108621, 138054, 175085, 221676, 280161, 353414, 445098, 559661
OFFSET
0,4
COMMENTS
A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers.
LINKS
The Unapologetic Mathematician weblog, Generalized Young Tableaux.
EXAMPLE
The a(8) = 19 generalized Young tableaux:
8 1 7 2 6 3 5 1 2 5 1 3 4
.
1 2 3 1 2 1 5 1 3 1 4 2 3 1 2 1 2 3
7 6 5 5 2 4 3 3 2 3 2
.
1 1 1 2
2 3 2
5 4 3
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]}]];
ptnplane[n_]:=Union[Map[primeMS, Join@@Permutations/@sqfacs[n], {2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y], And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])&&And@@(UnsameQ@@@DeleteCases[Transpose[PadRight[#]], 0, {2}])&]], {y, IntegerPartitions[n]}], {n, 10}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 16 2019
EXTENSIONS
a(21)-a(45) from Seiichi Manyama, Aug 19 2020
STATUS
approved