Each chef at "Sushi Emperor" prepares

Let \( n \) be the number of chefs and \( m \) be the number of customers at "Sushi Emperor" on Tuesday.

### Step 1: Determine total rolls prepared by chefs
Each chef prepares:
- \( 15 \) regular rolls
- \( 20 \) vegetarian rolls

Therefore, \( n \) chefs will prepare:
- Total regular rolls: \( 15n \)
- Total vegetarian rolls: \( 20n \)

### Step 2: Analyze the rolls eaten by customers
Each customer eats:
- \( 2 \) regular rolls
- \( 3 \) vegetarian rolls

Thus, \( m \) customers will eat:
- Total regular rolls eaten: \( 2m \)
- Total vegetarian rolls eaten: \( 3m \)

### Step 3: Analyze the remaining rolls
By the end of the day, it is stated that:
- \( 4 \) regular rolls remained uneaten
- \( 1 \) vegetarian roll remained uneaten

So, the total regular rolls after sales will be:
\[
15n - 2m = 4
\]
And the total vegetarian rolls remaining will be:
\[
20n - 3m = 1
\]

### Step 4: Set up the equations

We have the following system of equations:
1. \( 15n - 2m = 4 \) \quad (1)
2. \( 20n - 3m = 1 \) \quad (2)

### Step 5: Solve the equations

We can solve these equations using substitution or elimination. Let's eliminate \( m \).

From equation (1), we can express \( m \) in terms of \( n \):
\[
2m = 15n - 4 \\
m = \frac{15n - 4}{2} \quad \text{(3)}
\]

Substituting equation (3) into equation (2):
\[
20n - 3\left(\frac{15n - 4}{2}\right) = 1
\]
Multiply through by \( 2 \) to eliminate the fraction:
\[
40n - 3(15n - 4) = 2 \\
40n - 45n + 12 = 2 \\
-5n + 12 = 2
\]
Solving for \( n \):
\[
-5n = 2 - 12 \\
-5n = -10 \\
n = 2
\]

### Step 6: Substitute \( n \) back to find \( m \)

Substituting \( n = 2 \) back into equation (3):
\[
m = \frac{15(2) - 4}{2} = \frac{30 - 4}{2} = \frac{26}{2} = 13
\]

### Conclusion

The number of chefs is \( n = 2 \) and the number of customers is \( m = 13 \).

Thus, the final answer is:
There were \( 2 \) chefs and \( 13 \) customers.