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A323430 Number of rectangular plane partitions of n with strictly decreasing rows and columns.

%I A323430 #4 Jan 15 2019 09:24:36
%S A323430 1,1,1,3,3,5,7,9,12,16,22,27,36,44,57,72,89,110,139,170,210,261,318,
%T A323430 390,478,581,705,860,1036,1252,1511,1816,2178,2618,3127,3743,4471,
%U A323430 5330,6347,7564,8984,10674,12669,15016,17780,21050,24868,29371,34655,40836,48080
%N A323430 Number of rectangular plane partitions of n with strictly decreasing rows and columns.
%C A323430 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.
%e A323430 The a(8) = 12 matrices:
%e A323430   [8] [7 1] [6 2] [5 3] [5 2 1] [4 3 1]
%e A323430 .
%e A323430   [7] [6] [5] [3 2]
%e A323430   [1] [2] [3] [2 1]
%e A323430 .
%e A323430   [5] [4]
%e A323430   [2] [3]
%e A323430   [1] [1]
%e A323430 The a(10) = 22 matrices:
%e A323430   [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]
%e A323430 .
%e A323430   [9] [8] [7] [6] [5 2] [4 2] [4 3]
%e A323430   [1] [2] [3] [4] [2 1] [3 1] [2 1]
%e A323430 .
%e A323430   [7] [6] [5] [5]
%e A323430   [2] [3] [4] [3]
%e A323430   [1] [1] [1] [2]
%e A323430 .
%e A323430   [4]
%e A323430   [3]
%e A323430   [2]
%e A323430   [1]
%t A323430 Table[Sum[Length[Select[Union[Tuples[Select[IntegerPartitions[#,{k}],UnsameQ@@#&]&/@ptn]],And@@(OrderedQ[#,Greater]&/@Transpose[#])&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,30}]
%Y A323430 Cf. A000219, A003293, A047966, A101509, A114736, A117433, A299968, A319066.
%Y A323430 Cf. A323301, A323307, A323429, A323431, A323432, A323435, A323436, A323438.
%K A323430 nonn
%O A323430 0,4
%A A323430 _Gus Wiseman_, Jan 15 2019