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%I #21 Apr 01 2017 21:08:08
%S 1,3,1,6,1,2,12,1,4,3,20,1,4,5,5,35,1,6,8,9,7,54,1,6,10,12,11,11,86,1,
%T 8,13,20,14,19,15,128,1,8,18,23,22,25,23,22,192,1,10,21,34,30,37,29,
%U 36,30,275,1,10,26,41,41,48,41,45,46,42,399,1,12,32,56,53,72,52,67,58,66,56
%N Triangle read by rows: T(n,k) = sum of all parts that start in the k-th column of the diagram of regions of the set of partitions of n.
%C For the construction of the diagram see A225600.
%e For n = 5 and k = 3 the diagram of regions of the set of partitions of 5 contains three parts that start in the third column: two parts of size 1 and one part of size 2, therefore the sum of all parts that start in column 3 is 1 + 1 + 2 = 4, so T(5,3) = 4.
%e .
%e . Illustration of the parts
%e . Diagram that start in column k:
%e . for n=5 k=1 k=2 k=3 k=4 k=5
%e . _ _ _ _ _ _ _ _ _ _
%e . |_ _ _ | |_ _ _ _ _| _ _
%e . |_ _ _|_ | |_ _ _|_ |_ _| _
%e . |_ _ | | |_ _ _ _| _ _ |_|
%e . |_ _|_ | | |_ _|_ |_ _| _ |_|
%e . |_ _ | | | |_ _ _| _ |_| |_|
%e . |_ | | | | |_ _| _ |_| |_| |_|
%e . |_|_|_|_|_| |_| |_| |_| |_| |_|
%e .
%e k = 1 2 3 4 5
%e .
%e The 5th row lists: 20 1 4 5 5
%e .
%e Triangle begins:
%e 1;
%e 3, 1;
%e 6, 1, 2;
%e 12, 1, 4, 3;
%e 20, 1, 4, 5, 5;
%e 35, 1, 6, 8, 9, 7;
%e 54, 1, 6, 10, 12, 11, 11;
%e 86, 1, 8, 13, 20, 14, 19, 15;
%e 128, 1, 8, 18, 23, 22, 25, 23, 22;
%e 192, 1, 10, 21, 34, 30, 37, 29, 36, 30;
%e 275, 1, 10, 26, 41, 41, 48, 41, 45, 46, 42;
%e 399, 1, 12, 32, 56, 53, 72, 52, 67, 58, 66, 56;
%Y Column 1-2: A006128, A000012. Row sums give A066186. Right border gives A000041.
%Y Cf. A135010, A141285, A176572, A186114, A186412, A187219, A193870, A194446, A206437, A211978, A225598, A225600, A225610.
%K nonn,tabl
%O 1,2
%A _Omar E. Pol_, Aug 02 2013
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