A225598 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.

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

Offset: 1

Author

Omar E. Pol, Aug 02 2013

Keywords

• nonn
• tabf

Comments

For the definition of region see A206437.

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).

Examples

			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.
.
.    Diagram    Illustration of parts ending in column k:
.    for n=5      k=1   k=2     k=3       k=4        k=5
.   _ _ _ _ _                                  _ _ _ _ _
.  |_ _ _    |                _ _ _           |_ _ _ _ _|
.  |_ _ _|_  |               |_ _ _|  _ _ _ _       |_ _|
.  |_ _    | |          _ _          |_ _ _ _|        |_|
.  |_ _|_  | |         |_ _|  _ _ _      |_ _|        |_|
.  |_ _  | | |          _ _  |_ _ _|       |_|        |_|
.  |_  | | | |      _  |_ _|     |_|       |_|        |_|
.  |_|_|_|_|_|     |_|   |_|     |_|       |_|        |_|
.
k = 1 2 3 4 5
.
The 5th row lists:  1     5       8         9         12
.
Triangle begins:
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,  9, 17, 29, 24,  33, 25,  38;
1,  9, 23, 33, 36,  42, 39,  38, 49;
1, 11, 26, 47, 46,  61, 49,  61, 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;
		

Crossrefs

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.

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