G - Random Student ID
G - Random Student ID
Score : $600$ points
Problem Statement
Takahashi Elementary School has $N$ new students. For $i = 1, 2, \ldots, N$, the name of the $i$-th new student is $S_i$ (which is a string consisting of lowercase English letters). The names of the $N$ new students are distinct.
The $N$ students will be assigned a student ID $1, 2, 3, \ldots, N$ in ascending lexicographical order of their names. However, instead of the ordinary order of lowercase English letters where a is the minimum and z is the maximum, we use the following order:
- First, Principal Takahashi chooses a string $P$ from the $26!$ permutations of the string
abcdefghijklmnopqrstuvwxyzof length $26$, uniformly at random. - The lowercase English characters that occur earlier in $P$ are considered smaller.
For each of the $N$ students, find the expected value, modulo $998244353$, of the student ID assigned (see Notes).
What is the lexicographical order?
A string $S = S_1S_2\ldots S_{|S|}$ is said to be lexicographically smaller than a string $T = T_1T_2\ldots T_{|T|}$ if one of the following 1. and 2. holds. Here, $|S|$ and $|T|$ denote the lengths of $S$ and $T$, respectively.
- $|S| \lt |T|$ and $S_1S_2\ldots S_{|S|} = T_1T_2\ldots T_{|S|}$.
- There exists an integer $1 \leq i \leq \min\lbrace |S|, |T| \rbrace$ satisfying the following two conditions:
- $S_1S_2\ldots S_{i-1} = T_1T_2\ldots T_{i-1}$
- $S_i$ is a smaller character than $T_i$.
Notes
We can prove that the sought expected value is always a rational number. Moreover, under the Constraints of this problem, when the value is represented as $\frac{P}{Q}$ by two coprime integers $P$ and $Q$, we can prove that there is a unique integer $R$ such that $R \times Q \equiv P\pmod{998244353}$ and $0 \leq R \lt 998244353$. Find such $R$.
Constraints
- $2 \leq N$
- $N$ is an integer.
- $S_i$ is a string of length at least $1$ consisting of lowercase English letters.
- The sum of lengths of the given strings is at most $5 \times 10^5$.
- $i \neq j \Rightarrow S_i \neq S_j$
Input
Input is given from Standard Input in the following format:
Output
Print $N$ lines. For each $i = 1, 2, \ldots, N$, the $i$-th line should contain the expected value, modulo $998244353$, of the student ID assigned to Student $i$.
3
a
aa
ab
1
499122179
499122179
The expected value of the student ID assigned to Student $1$ is $1$; the expected values of the student ID assigned to Student $2$ and $3$ are $\frac{5}{2}$.
Note that the answer should be printed modulo $998244353$. For example, the sought expected value for Student $2$ and $3$ is $\frac{5}{2}$, and we have $2 \times 499122179 \equiv 5\pmod{998244353}$, so $499122179$ should be printed.
3
a
aa
aaa
1
2
3
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