Example 13 Let $n$ be a positive integer greater than 1, find the number of permutations $\left(a_{1}, a_{2}, \cdots, a_{n}\right)$ of $1,2, \cdots, n$ that satisfy the following property: there exists exactly one $i \in\{1,2, \cdots, n-1\}$, such that $a_{i} > a_{i+1}$.
Let $p_{n}$ denote the number of permutations with the given property. It is easy to see that $p_{1}=0, p_{2}=1$.
For $n \geqslant 2$, if $a_{n}=n$, then the number of such permutations is $p_{n-1}$;
If $a_{i}=n, 1 \leqslant i \leqslant n-1$, consider all such permutations. We can choose $i-1$ numbers from the $n-1$ numbers 1, $2, \cdots, n-1$ and arrange them in increasing order as $a_{1}, a_{2}, \cdots, a_{i-1}$, and arrange the remaining numbers in increasing order in the remaining positions. Thus, there are $\mathrm{C}_{n-1}^{i-1}$ ways to arrange them, so

that is,
$$
p_{n}=p_{n-1}+\sum_{i=1}^{n-1} \mathrm{C}_{n-1}^{i-1}=p_{n-1}+2^{n-1}-1 \text {, }
$$
$$
\begin{array}{c}
p_{n}-p_{n-1}=2^{n-1}-1, \\
p_{n-1}-p_{n-2}=2^{n-2}-1, \\
\cdots \cdots \cdots \\
p_{2}-p_{1}=2-1 .
\end{array}
$$

Adding up all these equations, we get
$$
\begin{aligned}
p_{n} & =\left(2^{n-1}-1\right)+\left(2^{n-2}-1\right)+\cdots+(2-1) \\
& =2^{n}-n-1 .
\end{aligned}
$$