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A225599 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.

%I A225599 #21 Apr 01 2017 21:08:08
%S A225599 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 A225599 8,13,20,14,19,15,128,1,8,18,23,22,25,23,22,192,1,10,21,34,30,37,29,
%U A225599 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 A225599 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 A225599 For the construction of the diagram see A225600.
%e A225599 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 A225599 .
%e A225599 .                       Illustration of the parts
%e A225599 .    Diagram             that start in column k:
%e A225599 .    for n=5       k=1          k=2  k=3    k=4    k=5
%e A225599 .   _ _ _ _ _       _ _ _ _ _
%e A225599 .  |_ _ _    |     |_ _ _ _ _|               _ _
%e A225599 .  |_ _ _|_  |     |_ _ _|_                 |_ _|   _
%e A225599 .  |_ _    | |     |_ _ _ _|          _ _          |_|
%e A225599 .  |_ _|_  | |     |_ _|_            |_ _|   _     |_|
%e A225599 .  |_ _  | | |     |_ _ _|            _     |_|    |_|
%e A225599 .  |_  | | | |     |_ _|         _   |_|    |_|    |_|
%e A225599 .  |_|_|_|_|_|     |_|          |_|  |_|    |_|    |_|
%e A225599 .
%e A225599 k = 1 2 3 4 5
%e A225599 .
%e A225599 The 5th row lists:  20           1    4      5      5
%e A225599 .
%e A225599 Triangle begins:
%e A225599 1;
%e A225599 3,   1;
%e A225599 6,   1,  2;
%e A225599 12,  1,  4,  3;
%e A225599 20,  1,  4,  5,  5;
%e A225599 35,  1,  6,  8,  9,  7;
%e A225599 54,  1,  6, 10, 12, 11, 11;
%e A225599 86,  1,  8, 13, 20, 14, 19, 15;
%e A225599 128, 1,  8, 18, 23, 22, 25, 23, 22;
%e A225599 192, 1, 10, 21, 34, 30, 37, 29, 36, 30;
%e A225599 275, 1, 10, 26, 41, 41, 48, 41, 45, 46, 42;
%e A225599 399, 1, 12, 32, 56, 53, 72, 52, 67, 58, 66, 56;
%Y A225599 Column 1-2: A006128, A000012. Row sums give A066186. Right border gives A000041.
%Y A225599 Cf. A135010, A141285, A176572, A186114, A186412, A187219, A193870, A194446, A206437, A211978, A225598, A225600, A225610.
%K A225599 nonn,tabl
%O A225599 1,2
%A A225599 _Omar E. Pol_, Aug 02 2013