(4) Let $n \geqslant 2$, for an $n$-tuple of ordered real numbers
$$
\begin{array}{c}
A=\left(a_{1}, a_{2}, \cdots, a_{n}\right) \\
b_{k}=\max _{1 \leqslant i \leqslant k} a_{i}, k=1,2, \cdots, n
\end{array}
$$

the array $B=\left(b_{1}, b_{2}, \cdots, b_{n}\right)$ is called the "innovation array" of $A$, and the number of distinct elements in $B$ is called the "innovation order" of $A$.

Consider all permutations of $1,2, \cdots, n$ (each permutation is regarded as an ordered array). For all permutations with an innovation order of 2, find the arithmetic mean of each term.
Let $a_{1}=m$, then $m$ can be any of $1, \cdots, n-1$.
When $a_{1}=m$, we call $m+1, \cdots, n$ “large numbers” and $1, \cdots, m-1$ “small numbers”. Among the “large numbers”, $n$ should be listed before the other numbers. Since there are $n-m$ “large numbers”, the number of permutations with an innovation index of 2 corresponding to $a_{1}=m$ is
$$
x_{m}=\mathrm{C}_{n-1}^{n-m}(n-m-1)!(m-1)!=\frac{(n-1)!}{n-m}
$$

Therefore, the arithmetic mean we are looking for is
$$
\begin{array}{c}
\frac{\sum_{m=1}^{n-1} m x_{m}}{\sum_{m=1}^{n-1} x_{m}}=\frac{(n-1)!\sum_{m=1}^{n-1} \frac{m}{n-m}}{(n-1)!\sum_{m=1}^{n-1} \frac{1}{n-m}}=\frac{\sum_{m=1}^{n-1} \frac{n-m}{m}}{\sum_{m=1}^{n-1} \frac{1}{m}}= \\
n-\frac{n-1}{1+\frac{1}{2}+\cdots+\frac{1}{n-1}}
\end{array}
$$