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A323450 Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are weakly increasing. 5
1, 1, 3, 6, 14, 26, 56, 103, 203, 374, 702, 1262, 2306, 4078, 7242, 12628, 21988, 37756, 64682, 109606, 185082, 309958, 516932, 856221, 1412461, 2316416, 3783552 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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(4) = 14 generalized Young tableaux:
4 1 3 2 2 1 1 2 1 1 1 1
.
1 2 1 1 1 2 1 1 1 1 1
3 2 2 1 1 1 1
.
1 1 1
1 1
2 1
.
1
1
1
1
The a(5) = 26 generalized Young tableaux:
5 1 4 2 3 1 1 3 1 2 2 1 1 1 2 1 1 1 1 1
.
1 2 1 1 1 3 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1
4 3 3 1 2 1 2 2 1 1 1 1
.
1 1 1 1 1 2 1 1 1 1 1
1 2 1 1 1 1 1
3 2 2 1 1 1
.
1 1 1
1 1
1 1
2 1
.
1
1
1
1
1
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[Sum[Length[Select[ptnplane[Times@@Prime/@y], And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])&]], {y, IntegerPartitions[n]}], {n, 10}]
CROSSREFS
Sequence in context: A049940 A265947 A358831 * A051749 A278788 A279828
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jan 16 2019
EXTENSIONS
a(16)-a(26) from Seiichi Manyama, Aug 19 2020
STATUS
approved

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Last modified June 13 03:07 EDT 2024. Contains 373366 sequences. (Running on oeis4.)