Let $\mathrm{dp}_{i,j,k}$ be “the number of ways to determine the $(k+1)$-th and later digits of $S_i,S_{i+1},\ldots,S_{j-1}$ such that $S_i \lt S_{i+1} \lt \ldots \lt S_{j-1}$ within those digits.” (DP stands for Dynamic Programming.) Here, within the specified segment, the $(k+1)$-th digits are weakly increasing; additionally, within a segment with the same $(k+1)$-th digits, the $(k+2)$-th and later digits must form weakly increasing integers. Thus, the DP can be evaluated by considering the ways to split $S_i,S_{i+1},\ldots,S_{j-1}$ into segments where the $(k+1)$-th segments are 0, 1, $\ldots$, and 9. Specifically, we evaluate $\mathrm{dp}_{i,j,k,l} =$ “the number of ways to determine the $(k+1)$-th and later digits of $S_i \lt S_{i+1} \lt \ldots \lt S_{j-1}$ such that $S_i \lt S_{i+1} \lt \ldots \lt S_{j-1}$ within those digits and zero or more first $(k+1)$-th digits are $l$,” which can be evaluated in a total of $\mathrm{O}(N^3Mb)$ time (where $b$ is the number of kinds of digits).