What is the number of partitions with distinct parts?

Such a partition is called a partition with distinct parts. If we count the partitions of 8 with distinct parts, we also obtain 6: This is a general property. For each positive number, the number of partitions with odd parts equals the number of partitions with distinct parts, denoted by q(n).

When is a partition of n a self conjugate partition?

Ex 3.3.7 A partition of n is self-conjugate if its Ferrers diagram is symmetric around the main diagonal, so that its conjugate is itself. Show that the number of self-conjugate partitions of n is equal to the number of partitions of n into distinct odd parts.

Are there any partitions that contain only odd parts?

Among the 22 partitions of the number 8, there are 6 that contain only odd parts : 7 + 1 5 + 3 5 + 1 + 1 + 1

Which is an example of a partition of an integer?

Partitions of integers have some interesting properties. Let p d (n) be the number of partitions of n into distinct parts; let p o (n) be the number of partitions into odd parts. Example 3.3.4 For n = 6, the partitions into distinct parts are 6, 5 + 1, 4 + 2, 3 + 2 + 1,

Are there any partitions that contain no 1s?

Some of these partitions contain no 1s, like 3+ 3+4+ 6, a partition of 16 into 4 parts. Subtracting 1 from each part, we get a partition of n− k into k parts; for the example, this is 2 +2+ 3+5. The remaining partitions of n into k parts contain a 1. If we remove the 1, we are left with a partition of n−1 into k −1 parts.

How to generate all unique partitions of an integer?

Input: n = 2 Output: 2 1 1 Input: n = 3 Output: 3 2 1 1 1 1 Note: 2+1 and 1+2 are considered as duplicates. Input: n = 4 Output: 4 3 1 2 2 2 1 1 1 1 1 1 Recommended: Please solve it on “ PRACTICE ” first, before moving on to the solution.

What are the Partitions of n into k parts?

Now consider the partitions of n into k parts. Some of these partitions contain no 1s, like 3+ 3+4+ 6, a partition of 16 into 4 parts. Subtracting 1 from each part, we get a partition of n− k into k parts; for the example, this is 2 +2+ 3+5. The remaining partitions of n into k parts contain a 1.