Procrastinate Counter

AtCoder

IDagc071_e

Time2000ms

Memory256MB

Difficulty—

For a permutation $P=(P_1,P_2,\dots,P_N)$ of $(1,2,\dots,N)$, define a length-$N$ integer sequence $f(P)$ as follows. * Initialize an integer sequence $X$ with $X=(P_1,P_2,\dots,P_N)$. Repeat the following operation until $X$ is a strictly increasing sequence. (It can be shown that $X$ will be a strictly increasing sequence in a finite number of operations for any permutation $P$.) * Let $k$ be the smallest index $i$ ($1 \leq i < N$) such that $X_i > X_{i+1}$. Move the $k$\-th element of $X$ to the end. More precisely, replace $X$ with $(X_1,X_2,\dots,X_{k-1},X_{k+1},X_{k+2},\dots,X_N,X_{k})$. Let $C_x$ be the number of times an integer $x$ is moved to the end during this series of operations. Then, define $f(P)=(C_1,C_2,\dots,C_N)$. Find the number, modulo a prime number $M$, of distinct sequences that can occur as $f(P)$. ## Constraints * $1 \leq N \leq 500$ * $10^8 \leq M \leq 10^9$ * $M$ is prime. * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ [samples]

Samples

Input #1

4 998244353

Output #1

8

For example, when $P=(1,4,3,2)$, we initialize $X=(1,4,3,2)$ and perform operations as follows:

1.  Since $X_1 \leq X_2$ but $X_2 > X_3$, the element $X_2=4$ is moved to the end, and $X$ becomes $(1,3,2,4)$.
2.  $X_2=3$ is moved to the end, and $X$ becomes $(1,2,4,3)$.
3.  $X_3=4$ is moved to the end, and $X$ becomes $(1,2,3,4)$.

The number of times $1,2,3,4$ are moved to the end is $0,0,1,2$, respectively, so $f(P)=(0,0,1,2)$.

Input #2

71 998244353

Output #2

473972314

Input #3

300 923223991

Output #3

333532467

API Response (JSON)

{
  "problem": {
    "name": "Procrastinate Counter",
    "description": {
      "content": "For a permutation $P=(P_1,P_2,\\dots,P_N)$ of $(1,2,\\dots,N)$, define a length-$N$ integer sequence $f(P)$ as follows. *   Initialize an integer sequence $X$ with $X=(P_1,P_2,\\dots,P_N)$. Repeat the f",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc071_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "For a permutation $P=(P_1,P_2,\\dots,P_N)$ of $(1,2,\\dots,N)$, define a length-$N$ integer sequence $f(P)$ as follows.\n\n*   Initialize an integer sequence $X$ with $X=(P_1,P_2,\\dots,P_N)$. Repeat the f...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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