Score : $600$ points

问题描述

Takahashi 小学有 $N$ 名新学生。对于 $i = 1, 2, \ldots, N$，第 $i$ 名新学生的姓名是 $S_i$（由小写英文字母组成的字符串）。这 $N$ 名新学生的姓名各不相同。

这 $N$ 名学生将根据他们的姓名的升序字典顺序被分配学号 $1, 2, 3, \ldots, N$。但是，我们不使用普通的字母表顺序（其中 a 是最小的，z 是最大的），而是采用以下顺序：

• 首先，高桥校长从长度为 $26$ 的字符串 abcdefghijklmnopqrstuvwxyz 的 $26!$ 种排列中，随机均匀选择一个字符串 $P$。
• 在 $P$ 中出现较早的小写英文字母被认为较小。

对于 $N$ 名学生中的每一位，计算并求出（见注释）所分配学号的期望值对 $998244353$ 取模的结果。

什么是字典顺序？

字符串 $S = S_1S_2\ldots S_{|S|}$ 被认为在字典序上小于字符串 $T = T_1T_2\ldots T_{|T|}$，当且仅当以下 1. 和 2. 中的一个条件成立。这里，$|S|$ 和 $|T|$ 分别表示字符串 $S$ 和 $T$ 的长度。

1. $|S| < |T|$ 并且 $S_1S_2\ldots S_{|S|} = T_1T_2\ldots T_{|S|}$。
2. 存在一个整数 $1 \leq i \leq \min\lbrace |S|, |T| \rbrace$ 满足以下两个条件：
   • $S_1S_2\ldots S_{i-1} = T_1T_2\ldots T_{i-1}$
   • $S_i$ 字符比 $T_i$ 字符更小。

以上为通义千问 qwen-max 翻译，仅供参考。

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 abcdefghijklmnopqrstuvwxyz of 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.

1. $|S| \lt |T|$ and $S_1S_2\ldots S_{|S|} = T_1T_2\ldots T_{|S|}$.
2. 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:

$N$

$S_1$

$S_2$

$\vdots$

$S_N$

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 $i$.

Sample Input 1

3
a
aa
ab

Sample Output 1

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.

Sample Input 2

3
a
aa
aaa

Sample Output 2

1
2
3

update @ 2024/3/10 11:20:00