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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. 4
1, 1, 3, 6, 14, 26, 56, 103, 203, 374, 702, 1262, 2306, 4078, 7242, 12628 (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

Table of n, a(n) for n=0..15.

nLab, Young Diagram.

The Unapologetic Mathematician weblog, Generalized Young Tableaux.

EXAMPLE

The a(4) = 14 generalized Young tableaux:

  4   13   22   112   1111

.

  1   2   11   12   11   111

  3   2   2    1    11   1

.

  1   11

  1   1

  2   1

.

  1

  1

  1

  1

The a(5) = 26 generalized Young tableaux:

  5   14   23   113   122   1112   11111

.

  1   2   11   13   12   11   111   112   111   1111

  4   3   3    1    2    12   2     1     11    1

.

  1   1   11   12   11   111

  1   2   1    1    11   1

  3   2   2    1    1    1

.

  1   11

  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

Cf. A000085, A000219, A003293, A053529, A114736, A138178, A296188, A299968.

Cf. A323436, A323437, A323438, A323439, A323451.

Sequence in context: A120940 A049940 A265947 * A051749 A278788 A279828

Adjacent sequences:  A323447 A323448 A323449 * A323451 A323452 A323453

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jan 16 2019

STATUS

approved

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Last modified January 18 01:05 EST 2020. Contains 330995 sequences. (Running on oeis4.)