Question

Prove that the number of partitions of a positive integer where each part occurs more than ones (1 s) is equal to the number of partitions into parts that either divisible by 2 or 3.

Answer 1

To prove that the number of partitions of a positive integer where each part occurs more than once is equal to the number of partitions into parts that either divisible by 2 or 3, we can use a bijection.

Step-by-step explanation:

To prove that the number of partitions of a positive integer where each part occurs more than once is equal to the number of partitions into parts that are divisible by 2 or 3, we can use a bijection. Let's consider a positive integer n.

1. If n is divisible by 2 or 3, we can divide it into parts that are divisible by 2 or 3 respectively. For example, if n = 6, we can partition it into {2, 2, 2} or {3, 3}.
2. If n is not divisible by 2 or 3, we can express it as a sum of parts where each part occurs more than once. For example, if n = 5, we can partition it into {2, 2, 1} or {3, 1, 1}.

Since both cases cover all possible partitions of n, we can conclude that the number of partitions in both cases are equal.