How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?
To solve the problem of finding the number of ways to line up 19 girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her, we can use mathematical induction.

1. **Base Case:**
   For \( n = 1 \), there is only one girl, and there is exactly 1 way to line her up. This satisfies the condition trivially.

2. **Inductive Step:**
   Assume that for \( n = k \) girls, there are \( 2^{k-1} \) ways to arrange them such that no girl has a shorter girl both in front of and behind her.

3. **Inductive Hypothesis:**
   We need to show that for \( n = k+1 \) girls, there are \( 2^k \) ways to arrange them under the same condition.

4. **Adding the \( (k+1) \)-th Girl:**
   Consider adding a new girl who is taller than all \( k \) existing girls. This new tallest girl can only be placed at either the front or the back of the line. If she is placed anywhere else, she would have a shorter girl both in front of and behind her, violating the condition.

5. **Positioning the Tallest Girl:**
   - If the tallest girl is placed at the front, the remaining \( k \) girls must be arranged in one of the \( 2^{k-1} \) valid ways.
   - If the tallest girl is placed at the back, the remaining \( k \) girls must again be arranged in one of the \( 2^{k-1} \) valid ways.

6. **Combining the Arrangements:**
   Since there are 2 choices for the position of the tallest girl (front or back), and each choice allows for \( 2^{k-1} \) valid arrangements of the remaining \( k \) girls, the total number of valid arrangements for \( k+1 \) girls is:
   \[
   2 \times 2^{k-1} = 2^k
   \]

By induction, the number of ways to arrange \( n \) girls such that no girl has a shorter girl both in front of and behind her is \( 2^{n-1} \).

7. **Conclusion for 19 Girls:**
   For \( n = 19 \), the number of ways to arrange the girls is:
   \[
   2^{19-1} = 2^{18} = 262144
   \]

The final answer is \( \boxed{ 262144 } \)