A323431 Number of strict rectangular plane partitions of n.

1, 1, 1, 3, 3, 5, 7, 9, 11, 15, 21, 25, 33, 41, 53, 65, 81, 97, 121, 143, 173, 215, 255, 305, 367, 441, 527, 637, 751, 899, 1067, 1269, 1491, 1775, 2071, 2439, 2875, 3357, 3911, 4577, 5309, 6177, 7171, 8305, 9609, 11151

Offset

0,4

Comments

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

Links

Table of n, a(n) for n=0..45.

Example

The a(10) = 21 matrices:
  [10] [9 1] [8 2] [7 3] [7 2 1] [6 4] [6 3 1] [5 4 1] [5 3 2] [4 3 2 1]
.
  [9] [8] [7] [6] [4 2] [4 3]
  [1] [2] [3] [4] [3 1] [2 1]
.
  [7] [6] [5] [5]
  [2] [3] [4] [3]
  [1] [1] [1] [2]
.
  [4]
  [3]
  [2]
  [1]

Mathematica

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

Crossrefs

Cf. A000219, A003293, A047966, A089299 A101509, A114736, A117433, A299968, A319066.

Cf. A323301, A323307, A323429, A323430, A323432, A323435, A323436, A323438.

Sequence in context: A323703 A333147 A323434 * A211516 A323430 A291941

Adjacent sequences: A323428 A323429 A323430 * A323432 A323433 A323434

Keyword

nonn

Author

Gus Wiseman, Jan 16 2019

Status

approved