A partition of a positive integer is even if all its elements are even numbers. Similarly, a partition
is odd if all its elements are odd. Determine all positive integers $n$ such that the number of even partitions of
$n$ is equal to the number of odd partitions of $n$.
Remark: A partition of a positive integer $n$ is a non-decreasing sequence of positive integers whose sum of
elements equals $n$. For example, $(2; 3; 4), (1; 2; 2; 2; 2)$ and $(9) $ are partitions of $9.$
To determine all positive integers \( n \) such that the number of even partitions of \( n \) is equal to the number of odd partitions of \( n \), we will analyze the structure of these partitions.

1. **Define Even and Odd Partitions:**
   - An even partition of \( n \) consists entirely of even numbers.
   - An odd partition of \( n \) consists entirely of odd numbers.

2. **Association of Partitions:**
   - Consider an even partition \((a_1, a_2, \ldots, a_k)\) where each \(a_i\) is even.
   - We can associate an odd partition to this even partition by transforming each even number \(a_i\) into \(a_i - 1\) and adding a 1 for each \(a_i\). This results in the partition \((1, 1, \ldots, 1, a_1 - 1, a_2 - 1, \ldots, a_k - 1)\).

3. **Conditions for Equal Number of Partitions:**
   - For the number of even partitions to equal the number of odd partitions, each odd partition must start with 1.
   - This implies that \( n \) must be even because if \( n \) were odd, there would be no even partitions.

4. **Checking Small Values of \( n \):**
   - For \( n = 2 \):
     - Even partitions: \((2)\)
     - Odd partitions: \((1, 1)\)
     - Both have 1 partition.
   - For \( n = 4 \):
     - Even partitions: \((4)\), \((2, 2)\)
     - Odd partitions: \((1, 1, 1, 1)\), \((1, 1, 2)\)
     - Both have 2 partitions.

5. **General Case Analysis:**
   - For \( n \geq 6 \):
     - Consider the odd partition \((3, n-3)\). This partition does not start with 1, which contradicts our requirement for equal numbers of partitions.
     - Therefore, \( n \geq 6 \) cannot satisfy the condition.

6. **Conclusion:**
   - The only values of \( n \) that satisfy the condition are \( n = 2 \) and \( n = 4 \).

The final answer is \( \boxed{ n = 2 } \) and \( n = 4 \).