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# [Solution] Multiset of Strings Codeforces Solution

F. Multiset of Strings
time limit per test
6 seconds
memory limit per test
512 megabytes
input
standard input
output
standard output

You are given three integers $n$$k$ and $f$.

Consider all binary strings (i. e. all strings consisting of characters $0$ and/or $1$) of length from $1$ to $n$. For every such string $s$, you need to choose an integer ${c}_{s}$ from $0$ to $k$.

A multiset of binary strings of length exactly $n$ is considered beautiful if for every binary string $s$ with length from $1$ to $n$, the number of strings in the multiset such that $s$ is their prefix is not exceeding ${c}_{s}$.

For example, let $n=2$${c}_{0}=3$${c}_{00}=1$${c}_{01}=2$${c}_{1}=1$${c}_{10}=2$, and ${c}_{11}=3$. The multiset of strings $\left\{11,01,00,01\right\}$ is beautiful, since:

• for the string $0$, there are $3$ strings in the multiset such that $0$ is their prefix, and $3\le {f}_{0}$;
• for the string $00$, there is one string in the multiset such that $00$ is its prefix, and $1\le {c}_{00}$;
• for the string $01$, there are $2$ strings in the multiset such that $01$ is their prefix, and $2\le {c}_{01}$;

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• for the string $1$, there is one string in the multiset such that $1$ is its prefix, and $1\le {c}_{1}$;
• for the string $10$, there are $0$ strings in the multiset such that $10$ is their prefix, and $0\le {c}_{10}$;
• for the string $11$, there is one string in the multiset such that $11$ is its prefix, and $1\le {c}_{11}$.

Now, for the problem itself. You have to calculate the number of ways to choose the integer ${c}_{s}$ for every

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Multiset of Strings Codeforces Solution

binary string $s$ of length from $1$ to $n$ in such a way that the maximum possible size of a beautiful multiset is exactly $f$.

Input

The only line of input contains three integers $n$$k$ and $f$ ($1\le n\le 15$$1\le k,f\le 2\cdot {10}^{5}$).

Output

Print one integer — the number of ways to choose the integer ${c}_{s}$ for every binary string $s$ of length from $1$ to $n$ in such a way that the maximum possible size of a beautiful multiset is exactly $f$. Since it can be huge, print it modulo $998244353$.