Tree and Intervals

AtCoder

IDagc069_d

Time4500ms

Memory256MB

Difficulty—

You are given an integer $N$ and a prime $P$. For a tree with $N$ vertices labeled $1$ through $N$, let $a_i$ and $b_i$ be the endpoints of the $i$\-th edge $(1 \leq i \leq N-1)$. Also, define $x_j\ (1 \leq j \leq N-1)$ as: * The number of integers $i\ (1 \leq i \leq N-1)$ satisfying $\min(a_i,b_i) \leq j \lt \max(a_i,b_i)$. Find the number, modulo $P$, of sequences that could be $(x_1, x_2, \ldots, x_{N - 1})$. ## Constraints * $2 \leq N \leq 500$ * $10^8 \leq P \leq 10^9$ * $P$ is prime. ## Input The input is given from Standard Input in the following format: $N$ $P$ [samples]

Samples

Input #1

3 998244353

Output #1

3

There are three different trees with three vertices, if we distinguish the vertices by their labels but not the edges.  
The corresponding $(x_1, x_2)$ are $(1, 1)$, $(2, 1)$, and $(1, 2)$, so the expected output is $3$ modulo $P = 998244353$.

Input #2

69 433416647

Output #2

243082757

Find the count modulo $P$.

API Response (JSON)

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      "content": "You are given an integer $N$ and a prime $P$.\nFor a tree with $N$ vertices labeled $1$ through $N$, let $a_i$ and $b_i$ be the endpoints of the $i$\\-th edge $(1 \\leq i \\leq N-1)$. Also, define $x_j\\ (...",
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