A225598 - OEIS

%I #18 Mar 13 2015 22:58:17

%S 1,1,3,1,3,5,1,5,5,9,1,5,8,9,12,1,7,11,15,12,20,1,7,14,19,19,20,25,1,

%T 9,17,29,24,33,25,38,1,9,23,33,36,42,39,38,49,1,11,26,47,46,61,49,61,

%U 49,69,1,11,32,55,63,76,70,76,76,69,87,1,13,38,73,78,110,87,111,95,108,87,123

%N Triangle read by rows: T(n,k) = sum of all parts of all regions of the set of partitions of n whose largest part is k.

%C For the definition of region see A206437.

%C T(n,k) is also the sum of all parts that end in the k-th column of the diagram of regions of the set of partitions of n (see Example section).

%e For n = 5 and k = 3 the set of partitions of 5 contains two regions whose largest part is 3, they are third region which contains three parts [3, 1, 1] and the sixth region which contains only one part [3]. Therefore the sum of all parts is 3 + 1 + 1 + 3 = 8, so T(5,3) = 8.

%e .

%e .    Diagram    Illustration of parts ending 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:  1     5       8         9         12

%e .

%e Triangle begins:

%e 1;

%e 1,  3;

%e 1,  3,  5;

%e 1,  5,  5,  9;

%e 1,  5,  8,  9, 12;

%e 1,  7, 11, 15, 12,  20;

%e 1,  7, 14, 19, 19,  20, 25;

%e 1,  9, 17, 29, 24,  33, 25,  38;

%e 1,  9, 23, 33, 36,  42, 39,  38, 49;

%e 1, 11, 26, 47, 46,  61, 49,  61, 49,  69;

%e 1, 11, 32, 55, 63,  76, 70,  76, 76,  69, 87;

%e 1, 13, 38, 73, 78, 110, 87, 111, 95, 108, 87, 123;

%Y Column 1 is A000012. Column 2 are the numbers >= 3 of A109613. Row sums give A066186. Right border gives A046746. Second right border gives A046746.

%Y Cf. A000041, A066186, A135010, A141285, A186114, A186412, A187219, A194446, A206437, A207779, A211978, A225597, A225600, A225610.

%K nonn,tabf

%O 1,3

%A _Omar E. Pol_, Aug 02 2013