A13356 | Strange Function

Let's define the function $f$ of multiset $a$ as the multiset of number of occurences of every number, that is present in $a$ .

E.g., $f(\{5, 5, 1, 2, 5, 2, 3, 3, 9, 5\}) = \{1, 1, 2, 2, 4\}$ .

Let's define $f^k(a)$ , as applying $f$ to array $a$ $k$ times: $f^k(a) = f(f^{k-1}(a)), f^0(a) = a$ .

E.g., $f^2(\{5, 5, 1, 2, 5, 2, 3, 3, 9, 5\}) = \{1, 2, 2\}$ .

You are given integers $n, k$ and you are asked how many different values the function $f^k(a)$ can have, where $a$ is arbitrary non-empty array with numbers of size no more than $n$ . Print the answer modulo $998\,244\,353$ .

The first and only line of input consists of two integers $n, k$ ( $1 \le n, k \le 2020$ ).

Print one number — the number of different values of function $f^k(a)$ on all possible non-empty arrays with no more than $n$ elements modulo $998\,244\,353$ .