103236: [Atcoder]ABC323 G - Inversion of Tree

Problem Statement

You are given a permutation $P=(P_1,P_2,\ldots,P_N)$ of $(1,2,\ldots,N)$.

For each $K=0,1,\ldots,N-1$, find the number, modulo $998244353$, of trees with $N$ vertices numbered $1$ to $N$ that satisfy the following condition.

• Among the pairs of vertices $(u_i,v_i)\ (u_i < v_i)$ that are directly connected by an edge in the tree, exactly $K$ pairs satisfy $P_{u_i}>P_{v_i}$.

Constraints

• $2\leq N\leq 500$
• $P$ is a permutation of $(1,2,\ldots,N)$.

Input

The input is given from Standard Input in the following format:

$N$ 
$P_1$ $P_2$ $\ldots$ $P_N$

Output

For each $K=0,1,\ldots,N-1$, print the number, modulo $998244353$, of trees that satisfy the condition, with spaces in between.

Sample Input 1

3
1 3 2

Sample Output 1

1 2 0

The answer for $K=0$ is $1$: the tree with edges connecting vertices $1,2$ and $1,3$. Indeed, $P_1 < P_2$ and $P_1<P_3$.

The answer for $K=1$ is $2$: the tree with edges connecting vertices $1,2$ and $2,3$, and another with edges connecting vertices $1,3$ and $2,3$. Indeed, in the tree with edges connecting vertices $1,2$ and $2,3$, we have $P_1 < P_2$, $P_2>P_3$.

Sample Input 2

10
3 1 4 10 8 6 9 2 7 5

Sample Output 2

294448 2989776 12112684 25422152 30002820 20184912 7484084 1397576 108908 2640

问题描述

给定一个排列$P=(P_1,P_2,\ldots,P_N)$，其中$(1,2,\ldots,N)$的排列。

对于每个$K=0,1,\ldots,N-1$，求满足以下条件的有$N$个顶点的树的数量，模$998244353$。

• 在树中直接由边连接的顶点对$(u_i,v_i)\ (u_i < v_i)$中，恰好有$K$对满足$P_{u_i}>P_{v_i}$。

约束条件

• $2\leq N\leq 500$
• $P$是$(1,2,\ldots,N)$的排列。

输入

输入从标准输入按以下格式给出：

$N$ 
$P_1$ $P_2$ $\ldots$ $P_N$

输出

对于每个$K=0,1,\ldots,N-1$，输出满足条件的树的数量，模$998244353$，之间用空格隔开。

样例输入1

3
1 3 2

样例输出1

1 2 0

对于$K=0$，答案为$1$：有边连接顶点$1,2$和$1,3$的树。确实，$P_1 < P_2$和$P_1<P_3$。

对于$K=1$，答案为$2$：有边连接顶点$1,2$和$2,3$的树，以及有边连接顶点$1,3$和$2,3$的树。确实，在有边连接顶点$1,2$和$2,3$的树中，我们有$P_1 < P_2$，$P_2>P_3$。

样例输入2

10
3 1 4 10 8 6 9 2 7 5

样例输出2

294448 2989776 12112684 25422152 30002820 20184912 7484084 1397576 108908 2640