A228716 Triangle read by rows in which row n lists the rows (including 0's) of the n-th section of the set of partitions (in colexicographic order) of any integer >= n.

1, 0, 1, 2, 0, 0, 1, 0, 1, 3, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 2, 4, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 3, 2, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 2, 2, 2, 4, 2, 3, 3, 6, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Author

Omar E. Pol, Sep 02 2013

Keywords

• nonn
• tabf

Comments

In other words, row n lists the rows of the last section of the set of partitions (in colexicographic order) of n.

Row lengths is A006128.

The number of zeros in row n is A006128(n-1).

Rows sums give A138879.

For more properties of the sections of the set of partitions of a positive integer see example.

Positive terms give A230440. - Omar E. Pol, Oct 25 2013

Examples

			Illustration of the 15 rows of the 7th section (including zeros) of the set of partitions of any integer >= 7 (hence this is also the last section of the set of partitions of 7). Note that the sum of the k-th column is equal to the number of parts >= k, therefore the first differences of the column sums give the number of occurrences of parts k in the section. The same for all sections of all positive integers, see below:
-----------------------------
Column: 1  2  3  4  5  6  7
-----------------------------
Row |
1   |   0, 0, 0, 0, 0, 0, 1;
2   |   0, 0, 0, 0, 0, 1;
3   |   0, 0, 0, 0, 1;
4   |   0, 0, 0, 0, 1;
5   |   0, 0, 0, 1;
6   |   0, 0, 0, 1;
7   |   0, 0, 1;
8   |   0, 0, 0, 1;
9   |   0, 0, 1;
10  |   0, 0, 1;
11  |   0, 1;
12  |   3, 2, 2;
13  |   5, 2;
14  |   4, 3;
15  |   7;
-----------------------------
Sums:  19, 8, 5, 3, 2, 1, 1 -> Row 7 of triangle A207031.
.       | /| /| /| /| /| /|
.       |/ |/ |/ |/ |/ |/ |
F.Dif: 11, 3, 2, 1, 1, 0, 1 -> Row 7 of triangle A182703.
.
Triangle begins:
[1];
[0,1],[2];
[0,0,1],[0,1],[3];
[0,0,0,1],[0,0,1],[0,1],[2,2],[4];
[0,0,0,0,1],[0,0,0,1],[0,0,1],[0,0,1],[0,1],[3,2],[5];
[0,0,0,0,0,1],[0,0,0,0,1],[0,0,0,1],[0,0,0,1],[0,0,1],[0,0,1],[0,1],[2,2,2],[4,2],[3,3],[6];
[0,0,0,0,0,0,1],[0,0,0,0,0,1],[0,0,0,0,1],[0,0,0,0,1],[0,0,0,1],[0,0,0,1],[0,0,1],[0,0,0,1],[0,0,1],[0,0,1],[0,1],[3,2,2],[5,2],[4,3],[7];
		

Crossrefs

Cf. A000041, A006128, A135010, A138121, A138879, A182703, A187219, A207031, A211992.