Let $S = \{1, 22, 333, \dots , 999999999\}$. For how many pairs of integers $(a, b)$ where $a, b \in S$ and $a < b$ is it the case that $a$ divides $b$?
To solve the problem, we need to determine how many pairs \((a, b)\) exist such that \(a, b \in S\) and \(a < b\) where \(a\) divides \(b\). The set \(S\) is given as \(\{1, 22, 333, \dots, 999999999\}\). For simplicity, we can represent each element in \(S\) as \(n\) repeated \(n\) times, where \(n\) ranges from 1 to 9.

1. **Identify the elements of \(S\):**
   \[
   S = \{1, 22, 333, 4444, 55555, 666666, 7777777, 88888888, 999999999\}
   \]

2. **Determine the pairs \((a, b)\) such that \(a < b\) and \(a\) divides \(b\):**
   - For \(b = 999999999\) (9 repeated 9 times):
     \[
     a = 1, 333, \Rightarrow 2 \text{ pairs}
     \]
   - For \(b = 88888888\) (8 repeated 8 times):
     \[
     a = 1, 22, 4444 \Rightarrow 3 \text{ pairs}
     \]
   - For \(b = 7777777\) (7 repeated 7 times):
     \[
     a = 1 \Rightarrow 1 \text{ pair}
     \]
   - For \(b = 666666\) (6 repeated 6 times):
     \[
     a = 1, 22, 333 \Rightarrow 3 \text{ pairs}
     \]
   - For \(b = 55555\) (5 repeated 5 times):
     \[
     a = 1 \Rightarrow 1 \text{ pair}
     \]
   - For \(b = 4444\) (4 repeated 4 times):
     \[
     a = 1, 22 \Rightarrow 2 \text{ pairs}
     \]
   - For \(b = 333\) (3 repeated 3 times):
     \[
     a = 1 \Rightarrow 1 \text{ pair}
     \]
   - For \(b = 22\) (2 repeated 2 times):
     \[
     a = 1 \Rightarrow 1 \text{ pair}
     \]

3. **Sum the number of valid pairs:**
   \[
   2 + 3 + 1 + 3 + 1 + 2 + 1 + 1 = 14
   \]

Conclusion:
\[
\boxed{14}
\]