WOLFRAM|DEMONSTRATIONS PROJECT

The Number of Partitions with a Given Largest Part

​
diagonal
1
​
largest part k
partition n
1
2
3
4
5
6
7
8
9
10
11
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14
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16
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18
19
20
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
9
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
10
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
11
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
12
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
13
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
14
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
15
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
16
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
17
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
18
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
19
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
20
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
The number
P(n,k)
is the number of ways of partitioning
n
into parts with largest part
k
. This is the same as partitioning
n
into exactly
k
parts. The function satisfies the recurrence relation
P(n,k)=P(n-1,k-1)+P(n-k,k)
. Move the slider to see the diagonal-by-diagonal construction of the table.