Sum of Hash of Lexmin

AtCoder

IDagc068_d

Time2000ms

Memory256MB

Difficulty—

There is a rooted tree $T$ with $N$ vertices numbered from $1$ to $N$. Vertex $1$ is the root, and the parent of vertex $i$ ($2 \leq i \leq N$) is $P_i$ ($P_i < i$). A permutation $x = (x_1, x_2, \cdots, x_N)$ of $(1, 2, \cdots, N)$ is judged to be a **good** permutation or not by the following criteria. * Consider the following operation on $x$. * Choose two adjacent elements $u$ and $v$ in $x$ such that $u$ and $v$ are in an ancestor-descendant relationship in $T$. It does not matter which is the ancestor and which is the descendant. Then, swap $u$ and $v$. * If it is possible to obtain a permutation that is lexicographically strictly smaller than the initial state by performing the above operation zero or more times, $x$ is **not** a good permutation. If it is impossible to obtain a permutation lexicographically smaller than the initial state by any such operations, $x$ is a good permutation. You are given a positive integer $B$. For a permutation $x$, define $\operatorname{hash}(x) = \sum_{1 \leq i \leq N} B^{i-1} \times x_i$. Find the sum of $\operatorname{hash}(x)$ over all good permutations $x$, modulo $998244353$. What is lexicographical order on sequences?A sequence $S = (S_1, S_2, \ldots, S_{|S|})$ is said to be **lexicographically smaller** than a sequence $T = (T_1, T_2, \ldots, T_{|T|})$ if and only if 1. or 2. below holds. Here, $|S|$ and $|T|$ denote the lengths of $S$ and $T$, respectively. 1. $|S| \lt |T|$ and $(S_1, S_2, \ldots, S_{|S|}) = (T_1, T_2, \ldots, T_{|S|})$. 2. There exists an integer $1 \leq i \leq \min{ |S|, |T| }$ such that the following two statements hold. * $(S_1, S_2, \ldots, S_{i-1}) = (T_1, T_2, \ldots, T_{i-1})$. * $S_i$ is smaller than $T_i$ (as a number). ## Constraints * $2 \leq N \leq 100$ * $1 \leq B < 998244353$ * $1 \leq P_i < i$ ($2 \leq i \leq N$) * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $B$ $P_2$ $P_3$ $\cdots$ $P_N$ [samples]

Samples

Input #1

3 100
1 1

Output #1

50502

For example, $x = (3, 1, 2)$ is not a good permutation, because by swapping $3$ and $1$, which are in an ancestor-descendant relationship, we can obtain $(1, 3, 2)$, a lexicographically smaller permutation.
In this sample, the good permutations are $x = (1, 2, 3)$ and $x = (1, 3, 2)$. Thus, the answer is $\operatorname{hash}((1,2,3)) + \operatorname{hash}((1,3,2)) = 30201 + 20301 = 50502$.

Input #2

5 100
1 2 3 4

Output #2

504030201

In this sample, any two vertices are in an ancestor-descendant relationship. Therefore, the only good permutation is $x = (1, 2, 3, 4, 5)$.

Input #3

10 248730679
1 2 1 2 5 6 1 8 1

Output #3

856673861

Input #4

20 480124393
1 2 3 2 3 4 3 8 3 4 11 10 4 14 15 12 17 18 19

Output #4

488941820

API Response (JSON)

{
  "problem": {
    "name": "Sum of Hash of Lexmin",
    "description": {
      "content": "There is a rooted tree $T$ with $N$ vertices numbered from $1$ to $N$. Vertex $1$ is the root, and the parent of vertex $i$ ($2 \\leq i \\leq N$) is $P_i$ ($P_i < i$). A permutation $x = (x_1, x_2, \\cdo",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc068_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a rooted tree $T$ with $N$ vertices numbered from $1$ to $N$. Vertex $1$ is the root, and the parent of vertex $i$ ($2 \\leq i \\leq N$) is $P_i$ ($P_i < i$).\nA permutation $x = (x_1, x_2, \\cdo...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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