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A319730 Number T(n,k) of plane partitions of n into parts of exactly k sorts which are introduced in ascending order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A319730 #20 Feb 18 2025 08:11:03
%S A319730 1,0,1,0,3,2,0,6,11,3,0,13,48,33,5,0,24,165,212,75,7,0,48,573,1253,
%T A319730 798,172,11,0,86,1759,6114,6175,2284,326,15,0,160,5473,29573,45040,
%U A319730 25697,6198,631,22,0,282,16051,131488,289685,238516,86189,14519,1102,30
%N A319730 Number T(n,k) of plane partitions of n into parts of exactly k sorts which are introduced in ascending order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A319730 Alois P. Heinz, <a href="/A319730/b319730.txt">Rows n = 0..50, flattened</a>
%H A319730 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PlanePartition.html">Plane partition</a>
%H A319730 Wikipedia, <a href="https://en.wikipedia.org/wiki/Plane_partition">Plane partition</a>
%F A319730 T(n,k) = 1/k! * A319600(n,k).
%e A319730 Triangle T(n,k) begins:
%e A319730   1;
%e A319730   0,   1;
%e A319730   0,   3,     2;
%e A319730   0,   6,    11,      3;
%e A319730   0,  13,    48,     33,      5;
%e A319730   0,  24,   165,    212,     75,      7;
%e A319730   0,  48,   573,   1253,    798,    172,    11;
%e A319730   0,  86,  1759,   6114,   6175,   2284,   326,    15;
%e A319730   0, 160,  5473,  29573,  45040,  25697,  6198,   631,   22;
%e A319730   0, 282, 16051, 131488, 289685, 238516, 86189, 14519, 1102, 30;
%e A319730   ...
%Y A319730 Columns k=0-1 give: A000007, A000219 (for n>0).
%Y A319730 Main diagonal gives A000041.
%Y A319730 Row sums give A319731.
%Y A319730 T(2n,n) gives A319732.
%Y A319730 Cf. A256130, A319600.
%K A319730 nonn,tabl
%O A319730 0,5
%A A319730 _Alois P. Heinz_, Sep 26 2018