Lexicographic Matching

You are given integers $N$, $K$ and an array $B$ of length $N$. Count the number of arrays $A$, also of length $N$, which satisfy the following conditions: - $1 \le A_i \le K$ - The lexicographically largest$^{\dagger}$ subsequence$^{\ddagger}$ of $A[1, i]$ has a length of $B_i$. $A[1, i]$ represents the prefix of the array containing $i$ elements, i.e. $[A_1, A_2, \ldots, A_i]$. Since the an