Inversion of Tree

AtCoder

IDabc323_g

Time4000ms

Memory256MB

Difficulty—

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: $N$ $P_1$ $P_2$ $\ldots$ $P_N$ [samples]

Samples

Input #1

3
1 3 2

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 $P_1 < P_2$, $P_2>P_3$.

Input #2

10
3 1 4 10 8 6 9 2 7 5

Output #2

294448 2989776 12112684 25422152 30002820 20184912 7484084 1397576 108908 2640

API Response (JSON)

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    "name": "Inversion of Tree",
    "description": {
      "content": "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 ",
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  "statements": [
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      "statement_type": "Markdown",
      "content": "You are given a permutation $P=(P_1,P_2,\\ldots,P_N)$ of $(1,2,\\ldots,N)$.\nFor each $K=0,1,\\ldots,N-1$, find the number, modulo $998244353$, of trees with $N$ vertices numbered $1$ to $N$ that satisfy ...",
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      "language": "English"
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