Hamming-Distant Arrays

AtCoder

IDagc076_a

Time2000ms

Memory256MB

Difficulty—

You are given an integer $N$. For length-$N^2$ integer sequences $a$ and $b$ consisting of integers between $1$ and $N$ (inclusive), we define their distance $d(a,b)$ as follows: * $d(a,b)=$ "the number of indices $i$ ($1 \leq i \leq N^2$) such that $a_i \neq b_i$." Now, you will create $N$ length-$N^2$ integer sequences consisting of integers between $1$ and $N$, and denote them as $x_1,x_2,\cdots,x_N$ (their order also matters). Find the number, modulo $998244353$, of instances of $(x_1,x_2,\cdots,x_N)$ satisfying the following condition. * For any length-$N^2$ integer sequence $y$ consisting of integers between $1$ and $N$, there exists some $1 \leq i \leq N$ such that $d(x_i,y) \geq N^2-N$. ## Constraints * $1 \leq N \leq 50$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ [samples]

Samples

Input #1

Output #1

80

For example, $(x_1,x_2)=((1,1,2,2),(1,2,1,2))$ does not satisfy the condition, because if $y=(1,1,1,2)$, then $d(x_1,y)=1,d(x_2,y)=1$.
On the other hand, $(x_1,x_2)=((1,1,1,1),(2,2,2,2))$ satisfies the condition.
There are $80$ instances of $(x_1,x_2)$ that satisfy the condition.

Input #2

Output #2

597965565

Input #3

Output #3

241191911

API Response (JSON)

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