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Number of rectangular plane partitions of n.
17

%I #6 Jan 15 2019 09:24:28

%S 1,1,3,5,10,14,26,35,58,81,124,169,257,345,501,684,968,1304,1830,2452,

%T 3387,4541,6188,8257,11193,14865,19968,26481,35341,46674,62007,81611,

%U 107860,141602,186292,243800,319610,416984,544601,708690,922472,1197018,1553442

%N Number of rectangular plane partitions of n.

%C 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.

%e The a(5) = 14 matrices:

%e [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]

%e .

%e [4] [3] [2 1]

%e [1] [2] [1 1]

%e .

%e [3] [2]

%e [1] [2]

%e [1] [1]

%e .

%e [2]

%e [1]

%e [1]

%e [1]

%e .

%e [1]

%e [1]

%e [1]

%e [1]

%e [1]

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

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

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Jan 15 2019