【Example 7】Arrange the positive integers $1,2, \cdots, n$ in a row, such that except for the number at the far left, each one differs by 1 from one of the numbers to its left (it can be 1 greater or 1 less). How many different arrangements are possible?
Let the number at the far left be $j$. To meet the requirements of the problem, the numbers $j-1, j-2, \cdots, 1$ must be arranged in decreasing order, and the numbers $j+1, j+2, \cdots, n$ must be arranged in increasing order. Therefore, for a fixed $j$, the number of arrangements is exactly the permutation number of non-distinct elements $\binom{n-1}{j-1 \quad n-j}=C_{n-1}^{j-1}$. However, the number $j$ at the far left can be any one of $1,2, \cdots, n$. Therefore, to find the total number of arrangements, we should sum over $j$ according to the addition principle. Thus, there are
$$
C_{n-1}^{0}+C_{n-1}^{1}+\cdots+C_{n-1}^{n-1}=2^{n-1}
$$

different arrangements.
In the above solution, we used the combinatorial identity (*), which is one of many combinatorial identities.