Count the Functions

For a function $f$, define $f^1(x) = f(x)$ and $f^m(x) = f(f^{m - 1}(x))$ for all $m > 1$. You are given $2$ integers $N$ and $K$. Find the number of functions $f : \{1, 2, 3, \ldots, N + K\} \rightarrow \{1, 2, 3, \ldots, N + K\}$ satisfying the following conditions: - $f(x) = x$ for all $x > N$. - $f^N(x) > N$ for all $x$. - $f^N(x) \le f^N(x + 1)$ for all $1 \le x < N$. Since the answer may