Problem 390

10. 112 .

From the problem, we know
$$\begin{array}{l} S_{n+1}-S_{n} \\ =|n|+2|n-1|+\cdots+10|n-9|- \\ {[|n-1|+2|n-2|+\cdots+10|n-10|] } \\ =|n|+|n-1|+\cdots+|n-9|-10|n-10| . \end{array}$$

When $n \geqslant 10$, $S_{n+1}-S_{n}>0$, thus, $S_{n}$ is monotonically increasing; $\square$
When $n=0$, $S_{1}-S_{0}>0$;
When $n=6$, $S_{7}-S_{6}<0$.
Therefore, $S_{n}$ is monotonically decreasing in the interval $[1,7]$ and monotonically increasing in the interval $[7,+\infty)$.
Hence, the minimum value of $S_{n}$ is $S_{7}=112$.