Counting Problem

You are given an array $A$ of $N$ integers. Some of the elements in the array are missing and are represented by the number $-1$. You need to replace each missing element (i.e. each $-1$) with an integer from the range $[1, K]$ (both inclusive). For each possible valid array (after filling all the missing positions), let $f(A)$ denote the number of non-empty strictly increasing subsequences of th