A sequence $y_1, y_2, \ldots, y_k$ of real numbers is called $\textit{zigzag}$ if $k=1$, or if $y_2-y_1, y_3-y_2, \ldots, y_k-y_{k-1}$ are nonzero and alternate in sign. Let $X_1, X_2, \ldots, X_n$ be chosen independently from the uniform distribution on $[0,1]$. Let $a\left(X_1, X_2, \ldots, X_n\right)$ be the largest value of $k$ for which there exists an increasing sequence of integers $i_1, i_2, \ldots, i_k$ such that $X_{i_1}, X_{i_2}, \ldots X_{i_k}$ is zigzag. Find the expected value of $a\left(X_1, X_2, \ldots, X_n\right)$ for $n \geq 2$.
1. **Definition and Initial Setup**:
   - A sequence \( y_1, y_2, \ldots, y_k \) of real numbers is called \textit{zigzag} if \( k=1 \), or if \( y_2 - y_1, y_3 - y_2, \ldots, y_k - y_{k-1} \) are nonzero and alternate in sign.
   - Let \( X_1, X_2, \ldots, X_n \) be chosen independently from the uniform distribution on \([0,1]\).
   - Define \( a(X_1, X_2, \ldots, X_n) \) as the largest value of \( k \) for which there exists an increasing sequence of integers \( i_1, i_2, \ldots, i_k \) such that \( X_{i_1}, X_{i_2}, \ldots, X_{i_k} \) is zigzag.

2. **Identifying Zigzag Sub-sequence**:
   - The largest zigzag sub-sequence consists of local maxima and local minima, including \( X_1 \) and \( X_n \).
   - This is because between any \( X_{i_{k-1}} \) and \( X_{i_{k+1}} \) in a zigzag sub-sequence, there must be a local maximum or minimum in the original sequence.

3. **Counting Local Maxima and Minima**:
   - \( a(X_1, X_2, \ldots, X_n) = 2 + \sum_{i=2}^{n-1} \mathbf{1}[X_i \text{ is local maximum/minimum}] \).
   - Here, \( \mathbf{1} \) is the indicator function which is 1 if \( X_i \) is a local maximum or minimum, and 0 otherwise.

4. **Probability Calculation**:
   - The probability that \( X_i \) is a local maximum or minimum is \( \frac{2}{3} \).
   - This is because there are \( 3! = 6 \) possible orderings of \( X_{i-1}, X_i, X_{i+1} \):
     \[
     \begin{align*}
     X_{i-1} &\leq X_i \leq X_{i+1}, \\
     X_{i-1} &\leq X_{i+1} \leq X_i, \\
     X_i &\leq X_{i-1} \leq X_{i+1}, \\
     X_i &\leq X_{i+1} \leq X_{i-1}, \\
     X_{i+1} &\leq X_{i-1} \leq X_i, \\
     X_{i+1} &\leq X_i \leq X_{i-1}.
     \end{align*}
     \]
   - Out of these 6 orderings, exactly 4 make \( X_i \) a local maximum or minimum.

5. **Expected Value Calculation**:
   - The expected number of local maxima/minima among \( X_2, X_3, \ldots, X_{n-1} \) is \( (n-2) \cdot \frac{2}{3} \).
   - Adding the contributions from \( X_1 \) and \( X_n \), we get:
     \[
     \mathbf{E}[a(X_1, X_2, \ldots, X_n)] = 2 + (n-2) \cdot \frac{2}{3}.
     \]
   - Simplifying this expression:
     \[
     \mathbf{E}[a(X_1, X_2, \ldots, X_n)] = 2 + \frac{2(n-2)}{3} = 2 + \frac{2n - 4}{3} = \frac{6 + 2n - 4}{3} = \frac{2n + 2}{3}.
     \]

The final answer is \(\boxed{\frac{2n + 2}{3}}\).