Let $n$ be a fixed positive integer, and choose $n$ positive integers $a_1, \ldots , a_n$. Given a permutation $\pi$ on the first $n$ positive integers, let $S_{\pi}=\{i\mid \frac{a_i}{\pi(i)} \text{ is an integer}\}$. Let $N$ denote the number of distinct sets $S_{\pi}$ as $\pi$ ranges over all such permutations. Determine, in terms of $n$, the maximum value of $N$ over all possible values of $a_1, \ldots , a_n$.

[i]Proposed by James Lin.[/i]
To determine the maximum value of \( N \) over all possible values of \( a_1, \ldots, a_n \), we need to analyze the sets \( S_{\pi} \) for all permutations \( \pi \) of the first \( n \) positive integers. 

1. **Define the Set \( S_{\pi} \)**:
   For a given permutation \( \pi \), the set \( S_{\pi} \) is defined as:
   \[
   S_{\pi} = \{ i \mid \frac{a_i}{\pi(i)} \text{ is an integer} \}
   \]
   This means \( i \in S_{\pi} \) if and only if \( \pi(i) \) divides \( a_i \).

2. **Matrix Representation**:
   We can represent the divisibility condition using a 0-1 matrix \( A \) of size \( n \times n \), where the entry \( A_{ij} \) is 1 if \( j \) divides \( a_i \), and 0 otherwise. For example, if \( n = 7 \) and \( a_k = k \) for all \( k \), the matrix \( A \) is:
   \[
   \begin{bmatrix}
   1 & 1 & 1 & 1 & 1 & 1 & 1 \\
   0 & 1 & 0 & 1 & 0 & 1 & 0 \\
   0 & 0 & 1 & 0 & 0 & 1 & 0 \\
   0 & 0 & 0 & 1 & 0 & 0 & 0 \\
   0 & 0 & 0 & 0 & 1 & 0 & 0 \\
   0 & 0 & 0 & 0 & 0 & 1 & 0 \\
   0 & 0 & 0 & 0 & 0 & 0 & 1 \\
   \end{bmatrix}
   \]

3. **Upper Bound**:
   The maximum number of distinct sets \( S_{\pi} \) is bounded by the number of subsets of \( \{1, 2, \ldots, n\} \), which is \( 2^n \). However, not all subsets can be achieved as \( S_{\pi} \).

4. **Counting Non-Attainable Sets**:
   - **All Rows are Different**: If all rows of the matrix \( A \) are different, then the complement of any row is not attainable. This is because for any permutation \( \pi \), the set \( S_{\pi} \) must correspond to a transversal of the matrix, and the complement of a row cannot be formed by such a transversal.
   - **Two Rows are the Same**: If two rows are the same, let \( R \) be this common row. Any set \( S \) differing from the complement of \( R \) in at most one position is not attainable. This is because any achievable set shares at least two characters with \( R \), and thus cannot equal \( S \).

5. **Construction**:
   We claim that taking \( a_k = k \) for all \( k \) works. This construction ensures that the matrix \( A \) has the form shown above, where each row corresponds to the divisors of \( k \). This matrix allows us to achieve \( 2^n - n \) distinct sets \( S_{\pi} \).

6. **Conclusion**:
   The maximum value of \( N \) is \( 2^n - n \).

The final answer is \( \boxed{ 2^n - n } \).