Score : $650$ points

问题描述

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

对于每个 $K=0,1,\ldots,N-1$，找出满足以下条件的有 $N$ 个顶点（编号为 $1$ 到 $N$）的树的数量，计算结果对 $998244353$ 取模。

• 在树中直接通过边相连的顶点对 $(u_i,v_i)\ (u_i < v_i)$ 中，恰好有 $K$ 对满足 $P_{u_i}>P_{v_i}$。

以上为通义千问 qwen-max 翻译，仅供参考。

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

update @ 2024/3/10 01:45:30