A182285 Triangle read by rows: T(n,k) = sum of all parts in the k-th zone of the last section of the set of partitions of n.

1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 1, 6, 6, 6, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 7, 7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8, 8, 8, 8, 8, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Omar E. Pol, Apr 23 2012

Keywords

• nonn
• tabf

Comments

Row n lists A000041(n-1) 1's together with A002865(n) n's.

Examples

			Illustration of three arrangements of the last section of the set of partitions of 7 and the zone numbers:
--------------------------------------------------------
Zone \   a)                    b)                    c)
--------------------------------------------------------
15      (7)                   (7)       (. . . . . . 7)
14      (4+3)               (4+3)       (. . . 4 . . 3)
13      (5+2)               (5+2)       (. . . . 5 . 2)
12      (3+2+2)           (3+2+2)       (. . 3 . 2 . 2)
11        (1)                 (1)                   (1)
10          (1)               (1)                   (1)
9           (1)               (1)                   (1)
8             (1)             (1)                   (1)
7           (1)               (1)                   (1)
6             (1)             (1)                   (1)
5             (1)             (1)                   (1)
4               (1)           (1)                   (1)
3               (1)           (1)                   (1)
2                 (1)         (1)                   (1)
1                   (1)       (1)                   (1)
.
For n = 7 and k = 12 we can see that in the 12th zone of the last section of 7 the parts are 3, 2, 2, therefore T(7,12) = 3+2+2 = 7.
Written as a triangle begins:
1;
1,2;
1,1,3;
1,1,1,4,4;
1,1,1,1,1,5,5;
1,1,1,1,1,1,1,6,6,6,6;
1,1,1,1,1,1,1,1,1,1,1,7,7,7,7;
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,8,8,8,8,8,8,8;
		

Crossrefs

Row n has length A000041(n). Row sums give A138879.

Cf. A000041, A002865, A135010, A138121, A182284, A193173.