Let $ {{a} _ {1}}, {{a} _ {2}}, \ldots, {{a} _ {n}} $ be permutation of numbers $ 1,2, \ldots, n $, where $ n \geq 2 $.
Find the maximum value of the sum $$ S (n) = | {{a} _ {1}} - {{a} _ {2}} | + | {{a} _ {2}} - {{a} _ {3}} | + \cdots + | {{a} _ {n-1}} - {{a} _ {n}} |. $$
1. **Understanding the Problem:**
   We need to find the maximum value of the sum 
   \[
   S(n) = |a_1 - a_2| + |a_2 - a_3| + \cdots + |a_{n-1} - a_n|
   \]
   where \(a_1, a_2, \ldots, a_n\) is a permutation of the numbers \(1, 2, \ldots, n\).

2. **Analyzing the Absolute Differences:**
   To maximize \(S(n)\), we need to maximize each term \(|a_i - a_{i+1}|\). The maximum difference between any two numbers in the set \(\{1, 2, \ldots, n\}\) is \(n-1\).

3. **Constructing the Permutation:**
   Consider the permutation that alternates between the largest and smallest remaining numbers. For example, for \(n = 4\), the permutation would be \(1, 4, 2, 3\). This ensures that the differences are maximized.

4. **General Pattern:**
   For a general \(n\), the permutation can be constructed as follows:
   - Start with the smallest number, then the largest, then the second smallest, then the second largest, and so on.
   - This can be written as \(1, n, 2, n-1, 3, n-2, \ldots\).

5. **Calculating the Sum:**
   Let's calculate \(S(n)\) for this permutation:
   - For \(n = 4\), the permutation is \(1, 4, 2, 3\):
     \[
     S(4) = |1 - 4| + |4 - 2| + |2 - 3| = 3 + 2 + 1 = 6
     \]
   - For \(n = 5\), the permutation is \(1, 5, 2, 4, 3\):
     \[
     S(5) = |1 - 5| + |5 - 2| + |2 - 4| + |4 - 3| = 4 + 3 + 2 + 1 = 10
     \]

6. **General Formula:**
   The sum \(S(n)\) for the optimal permutation is:
   \[
   S(n) = (n-1) + (n-2) + \cdots + 1 = \sum_{k=1}^{n-1} k = \frac{(n-1)n}{2}
   \]

7. **Conclusion:**
   The maximum value of \(S(n)\) is given by:
   \[
   S(n) = \frac{(n-1)n}{2}
   \]

The final answer is \(\boxed{\frac{(n-1)n}{2}}\)