Triangle begins:
1;
2;
3, 1;
4, 2, 1;
5, 3, 2, 1, 1;
6, 4, 3, 2, 2, 1, 1;
7, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1;
8, 6, 5, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1;
9, 7, 6, 5, 5, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
...
For n = 6, by definition the length of row 6 is
A000041(6-1) =
A000041(5) = 7, so the row 6 of triangle has seven terms. Since every column lists the positive integers
A000027 so the row 6 is [6, 4, 3, 2, 2, 1, 1].
Then we have that the divisors of the numbers of the 6th row are:
.
6th row of the triangle ----------> 6 4 3 2 2 1 1
3 2 1 1 1
2 1
1
.
There are seven 1's, four 2's, two 3's, one 4 and one 6.
In total there are 7 + 4 + 2 + 1 + 1 = 15 divisors.
On the other hand the last section of the set of the partitions of 6 can be represented in several ways, five of them as shown below:
._ _ _ _ _ _
|_ _ _ | 6 6 6 6
|_ _ _|_ | 3 3 3 3 3 3 3 3
|_ _ | | 4 2 4 2 4 2 4 2
|_ _|_ _|_ | 2 2 2 2 2 2 2 2 2 2 2 2
| | 1 1 1 1
| | 1 1 1 1
| | 1 1 1 1
| | 1 1 1 1
| | 1 1 1 1
| | 1 1 1 1
|_| 1 1 1 1
.
Figure 1. Figure 2. Figure 3. Figure 4. Figure 5.
.
In every figure there are seven 1's, four 2's, two 3's, one 4 and one 6, as shown also the 6th row of
A182703.
In total there are 7 + 4 + 2 + 1 + 1 =
A138137(6) = 15 parts in every figure.
Figure 5 is an arrangement that shows the correspondence between divisors and parts since the columns give the divisors of the terms of 6th row of triangle.
Finally we can see that all divisors of all numbers in the 6th row of the triangle are the same positive integers as all parts in the last section of the set of the partitions of 6.
