A323429 Number of rectangular plane partitions of n.

1, 1, 3, 5, 10, 14, 26, 35, 58, 81, 124, 169, 257, 345, 501, 684, 968, 1304, 1830, 2452, 3387, 4541, 6188, 8257, 11193, 14865, 19968, 26481, 35341, 46674, 62007, 81611, 107860, 141602, 186292, 243800, 319610, 416984, 544601, 708690, 922472, 1197018, 1553442

Offset

0,3

Comments

Number of ways to fill a (not necessarily square) matrix with the parts of an integer partition of n so that the rows and columns are weakly decreasing.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

G.f.: 1 + Sum_{k>=1, l>=1} q^(k*l) * Product_{i=1..k} Product_{j=1..l} 1/(1 - q^(i+j-1)). - Seiichi Manyama, May 10 2026

Example

The a(5) = 14 matrices:
  [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
.
  [4] [3] [2 1]
  [1] [2] [1 1]
.
  [3] [2]
  [1] [2]
  [1] [1]
.
  [2]
  [1]
  [1]
  [1]
.
  [1]
  [1]
  [1]
  [1]
  [1]

Mathematica

Table[Sum[Length[Select[Union[Sort/@Tuples[IntegerPartitions[#, {k}]&/@ptn]], And@@OrderedQ/@Transpose[#]&]], {ptn, IntegerPartitions[n]}, {k, Min[ptn]}], {n, 30}]

Prog

(PARI) my(N=50, q='q+O('q^N)); Vec(1+sum(k=1, N, sum(l=1, N\k, q^(k*l)*prod(i=1, k, prod(j=1, l, 1/(1-q^(i+j-1))))))) \\ Seiichi Manyama, May 10 2026

Crossrefs

Row sums of A395923.

Cf. A000219, A003293, A047966, A101509, A114736, A117433, A299968, A319066.

Cf. A323301, A323307, A323430, A323431, A323432, A323435, A323436, A323438.

Sequence in context: A220489 A229915 A092269 * A319066 A306319 A182722

Adjacent sequences: A323426 A323427 A323428 * A323430 A323431 A323432

Keyword

nonn

Author

Gus Wiseman, Jan 15 2019

Status

approved