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
Seiichi Manyama, Table of n, a(n) for n = 0..50
nLab, Young Diagram.
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 16 2019
EXTENSIONS
a(21)-a(45) from Seiichi Manyama, Aug 19 2020
STATUS
approved