7.2 can be represented as $2=2=1+1, 3$ can be represented as $3=3=1+2=2+1=1+1+1, 4$ can be represented as $4=4=1+3=3+1=2+2=2+1+1=1+2+1=1+1+2=1+1+1+1$ and so on. The above method of representing positive integers is called the ordered partition of positive integers. Find the number of ordered partitions of a natural number $n$.


7. Let the set of ordered partitions of a natural number $n$ be $S$. Now take $n$ identical balls and arrange them in a row, which can form $n-1$ gaps. Now choose $k(0 \leqslant k \leqslant n-1)$ out of the $n-1$ gaps, and insert a "board" in each of the selected gaps, thus dividing all the balls into $k+1$ groups. Denote the number of balls in each group as $x_{1}, x_{2}, x_{3}, \cdots, x_{k+1}$, then $x_{1}+x_{2}+\cdots+x_{k+1}=n$, which is an ordered partition of $n$. It is easy to see that different ways of inserting "boards" correspond 1-to-1 to different ordered partitions. Since there are 2 choices (not inserting or inserting) for each of the $n-1$ gaps, we have $|S|=2^{n-1}$.