Count Sorted Arrays

AtCoder

IDarc160_f

Time2000ms

Memory256MB

Difficulty—

There are an integer $N$ and $M$ pairs of integers: $(a_1, b_1), (a_2, b_2), \dots, (a_M, b_M)$. Each pair $(a_i, b_i)$ satisfies $1 \leq a_i \lt b_i \leq N$. Initally, you have all $N!$ permutations of $(1,2,\dots,N)$. You will perform $M$ operations. The $i$\-th operation is as follows. * Do the following for each of your $N!$ permutations. * Compare the $a_i$\-th and $b_i$\-th elements. If the former is greater, swap the two elements. For each $1 \leq i \leq M$, let $S_i$ be the number of permutations of yours that are already sorted in ascending order at the end of the $i$\-th operation. Print $S_1, S_2, \dots, S_M$. Here, the input gives you pairs of integers $(x_i, y_i)$ instead of $(a_i, b_i)$. The values of $(a_i, b_i)$ can be obtained using $x_i$, $y_i$, and $S_{i-1}$ as follows. (Let $S_0 = 1$ for convenience.) * Let $c_i = ((x_i + S_{i-1}) \bmod N) + 1$. * Let $d_i = ((y_i + S_{i-1} \times 2) \bmod N) + 1$. (Here, it is guaranteed that $c_i \neq d_i$.) * Let $a_i = \min(c_i, d_i)$. * Let $b_i = \max(c_i, d_i)$. ## Constraints * $2 \leq N \leq 15$ * $1 \leq M \leq 5 \times 10^5$ * $1 \leq a_i \lt b_i \leq N$ * $0 \leq x_i, y_i \leq N - 1$ ## Input The input is given from Standard Input in the following format: $N$ $M$ $x_1$ $y_1$ $x_2$ $y_2$ $\vdots$ $x_M$ $y_M$ [samples]

Samples

Input #1

2 1
1 1

Output #1

2

You start with the permutations $(1, 2)$ and $(2, 1)$.  
We have $(a_1, b_1) = (1, 2)$. At the end of the first operation, you have two copies of $(1, 2)$, so you should print $2$.

Input #2

Output #2

2
4
4
6

$(a_i, b_i)$ in order are $(1, 2), (2, 3), (1, 3), (1, 2)$.

Input #3

Output #3

2
4
4
8
16

$(a_i, b_i)$ in order are $(1, 2), (3, 4), (1, 5), (2, 3), (4, 5)$.

API Response (JSON)

{
  "problem": {
    "name": "Count Sorted Arrays",
    "description": {
      "content": "There are an integer $N$ and $M$ pairs of integers: $(a_1, b_1), (a_2, b_2), \\dots, (a_M, b_M)$. Each pair $(a_i, b_i)$ satisfies $1 \\leq a_i \\lt b_i \\leq N$. Initally, you have all $N!$ permutations ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc160_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are an integer $N$ and $M$ pairs of integers: $(a_1, b_1), (a_2, b_2), \\dots, (a_M, b_M)$. Each pair $(a_i, b_i)$ satisfies $1 \\leq a_i \\lt b_i \\leq N$.\nInitally, you have all $N!$ permutations ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments