To solve this problem, we need to examine the conditions given for the pairs $(a, b)$ of positive integers:
1. The greatest common divisor (GCD) of $a$ and $b$ is $1 \cdot 2 \cdot 3 \cdot \ldots \cdot 50$.
2. The least common multiple (LCM) of $a$ and $b$ is $1^2 \cdot 2^2 \cdot 3^2 \cdot \ldots \cdot 50^2$.

### Step 1: Express Conditions Using Prime Factorization

First, let's write both the GCD and LCM conditions using prime factorization.

Let $P = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot 50$.
The prime factorization of $P$ would be:
$$P = 2^{p_2} \cdot 3^{p_3} \cdot 5^{p_5} \cdot \ldots \cdot 47^{p_{47}}$$
where $p_k$ is the power of the prime $k$ in the factorization of the numbers from $1$ to $50$.

The LCM condition translates to:
$$\text{LCM}(a, b) = 1^2 \cdot 2^2 \cdot 3^2 \cdot \ldots \cdot 50^2 = 2^{2p_2} \cdot 3^{2p_3} \cdot 5^{2p_5} \cdot \ldots \cdot 47^{2p_{47}}$$

### Step 2: Relate GCD and LCM to $(a, b)$

For $(a, b)$, we have:
- $a = 2^{a_2} \cdot 3^{a_3} \cdot \ldots \cdot 47^{a_{47}}$
- $b = 2^{b_2} \cdot 3^{b_3} \cdot \ldots \cdot 47^{b_{47}}$

For each prime $k$:
$$\text{GCD}(a, b) = k^{\min(a_k, b_k)} = k^{p_k}$$
$$\text{LCM}(a, b) = k^{\max(a_k, b_k)} = k^{2p_k}$$

Thus, we have:
- $\min(a_k, b_k) = p_k$
- $\max(a_k, b_k) = 2p_k$

### Step 3: Determine the Number of Solutions for Each Prime

For each prime $k$:
- If $a_k = p_k$, then $b_k$ can be any integer such that $b_k \geq p_k$ and $b_k \leq 2p_k$.
- Similarly, if $b_k = p_k$, $a_k$ can be any integer such that $a_k \geq p_k$ and $a_k \leq 2p_k$.

This provides two options for each prime $k$:
1. $(a_k, b_k) = (p_k, 2p_k)$
2. $(a_k, b_k) = (2p_k, p_k)$

Since each prime can independently take one of these configurations, we have $2$ choices per prime. Given there are $15$ prime numbers from $1$ to $50$ (including repeated primes in products like $2^4$, $3^3$, etc.), the total number of pairs $(a, b)$ is:
$$2^{15} = 32768$$

Thus, the number of pairs $(a, b)$ satisfying the given conditions is:
$$\boxed{32768}$$