Creative Splitting

AtCoder

IDagc055_f

Time4000ms

Memory256MB

Difficulty—

You are given integers $N$ and $K$. Let's call an integer array $a=[a_1, a_2, \ldots, a_K]$ of length $K$ **good**, if $1 \le a_i \le i$ for all $1 \le i \le K$. Let's call an integer array $b=[b_1, b_2, \ldots, b_{NK}]$ of length $NK$ **amazing**, if it can be split into $N$ (not necessarily contiguous) subsequences of length $K$, each of which is good. Let's define $f(pos, val)$ as the number of amazing sequences $b$ such that $b_{pos}=val$. Find $f(pos, val)$ for all $1\le pos \le NK$, $1 \le val \le K$. As these numbers can be very big, output them modulo some prime $P$. ## Constraints * $1 \le N \le 20$. * $1 \le K \le 20$. * $10^8 \le P \le 10^9$ * $P$ is a prime. ## Input Input is given from Standard Input in the following format: $N$ $K$ $P$ [samples]

Samples

Input #1

2 2 965166677

Output #1

6 0 
4 2 
4 2 
3 3 

There exist $6$ amazing arrays:

*   $[1, 1, 1, 1]$ ー can be split into $[b_1, b_2], [b_3, b_4]$.
*   $[1, 1, 1, 2]$ ー can be split into $[b_1, b_2], [b_3, b_4]$.
*   $[1, 2, 1, 1]$ ー can be split into $[b_1, b_2], [b_3, b_4]$.
*   $[1, 2, 1, 2]$ ー can be split into $[b_1, b_2], [b_3, b_4]$.
*   $[1, 1, 2, 1]$ ー can be split into $[b_1, b_3], [b_2, b_4]$.
*   $[1, 1, 2, 2]$ ー can be split into $[b_1, b_3], [b_2, b_4]$.

API Response (JSON)

{
  "problem": {
    "name": "Creative Splitting",
    "description": {
      "content": "You are given integers $N$ and $K$. Let's call an integer array $a=[a_1, a_2, \\ldots, a_K]$ of length $K$ **good**, if $1 \\le a_i \\le i$ for all $1 \\le i \\le K$. Let's call an integer array $b=[b_1, b",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 4000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc055_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given integers $N$ and $K$.\nLet's call an integer array $a=[a_1, a_2, \\ldots, a_K]$ of length $K$ **good**, if $1 \\le a_i \\le i$ for all $1 \\le i \\le K$.\nLet's call an integer array $b=[b_1, b...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments