Subset MEX Counting

For a given integer array $A$ of length $N$, and satisfying $0 \le A_i < N$, and another integer parameter $M$ ($1 \le M \le N$), define $f(A, M)$ in the following way: - Start with an empty set $S = \{\}$. - For each $M$ length subsequence of $A$, i.e. all sequences of the form $[A_{i_1}, A_{i_2}, \ldots, A_{i_M}]$ with $1 \le i_1 < i_2 < \ldots < i_M \le N$, insert the MEX$^{\dagger}$ of the su