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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.
2
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
1,4
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
EXAMPLE
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
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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];
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Sep 02 2013
STATUS
approved