[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$n$, 𝑘$k$ and 𝑓$f$.

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

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

For example, let 𝑛=2$n=2$, 𝑐0=3$c_0=3$, 𝑐00=1$c_{00}=1$, 𝑐01=2$c_{01}=2$, 𝑐1=1$c_1=1$, 𝑐10=2$c_{10}=2$, and 𝑐11=3$c_{11}=3$. The multiset of strings {11,01,00,01}$\{11,01,00,01\}$ is beautiful, since:

• for the string 0$0$, there are 3$3$ strings in the multiset such that 0$0$ is their prefix, and 3≤𝑓0$3\le f_0$;
• for the string 00$00$, there is one string in the multiset such that 00$00$ is its prefix, and 1≤𝑐00$1\le c_{00}$;
• for the string 01$01$, there are 2$2$ strings in the multiset such that 01$01$ is their prefix, and 2≤𝑐01$2\le c_{01}$;

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• for the string 1$1$, there is one string in the multiset such that 1$1$ is its prefix, and 1≤𝑐1$1\le c_1$;
• for the string 10$10$, there are 0$0$ strings in the multiset such that 10$10$ is their prefix, and 0≤𝑐10$0\le c_{10}$;
• for the string 11$11$, there is one string in the multiset such that 11$11$ is its prefix, and 1≤𝑐11$1\le c_{11}$.

Now, for the problem itself. You have to calculate the number of ways to choose the integer 𝑐𝑠$c_s$ for every

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Multiset of Strings Codeforces Solution

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

Input

The only line of input contains three integers 𝑛$n$, 𝑘$k$ and 𝑓$f$ (1≤𝑛≤15$1\le n\le 15$; 1≤𝑘,𝑓≤2⋅105$1\le k,f\le 2\cdot {10}^5$).

Output

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