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A323430
Number of rectangular plane partitions of n with strictly decreasing rows and columns.
7
1, 1, 1, 3, 3, 5, 7, 9, 12, 16, 22, 27, 36, 44, 57, 72, 89, 110, 139, 170, 210, 261, 318, 390, 478, 581, 705, 860, 1036, 1252, 1511, 1816, 2178, 2618, 3127, 3743, 4471, 5330, 6347, 7564, 8984, 10674, 12669, 15016, 17780, 21050, 24868, 29371, 34655, 40836, 48080
OFFSET
0,4
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 strictly decreasing.
EXAMPLE
The a(8) = 12 matrices:
[8] [7 1] [6 2] [5 3] [5 2 1] [4 3 1]
.
[7] [6] [5] [3 2]
[1] [2] [3] [2 1]
.
[5] [4]
[2] [3]
[1] [1]
The a(10) = 22 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] [5 2] [4 2] [4 3]
[1] [2] [3] [4] [2 1] [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[Tuples[Select[IntegerPartitions[#, {k}], UnsameQ@@#&]&/@ptn]], And@@(OrderedQ[#, Greater]&/@Transpose[#])&]], {ptn, IntegerPartitions[n]}, {k, Min[ptn]}], {n, 30}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 15 2019
STATUS
approved