Avoid Permutations

AtCoder

IDarc118_e

Time2000ms

Memory256MB

Difficulty—

Let us consider the problem below for a permutation $P = (P_1, P_2, \ldots, P_N)$ of $(1, 2, \ldots, N)$, and denote the answer by $f(P)$. > We have an $(N+2)\times (N+2)$ grid, where the rows are numbered $0, 1, \ldots, N+1$ from top to bottom and the columns are numbered $0, 1, \ldots, N+1$ from left to right. Let $(i,j)$ denote the square at the $i$\-th row and $j$\-th column. > Consider paths from $(0, 0)$ to $(N+1, N+1)$ where we repeatedly move one square right or one square down. However, for each $1\leq i\leq N$, we must not visit the square $(i, P_i)$. How many such paths are there? You are given a positive integer $N$ and an integer sequence $A = (A_1, A_2, \ldots, A_N)$, where each $A_i$ is $-1$ or an integer between $1$ and $N$ (inclusive). Consider all permutations $P = (P_1, P_2, \ldots, P_N)$ of $(1, 2, \ldots, N)$ satisfying $A_i\neq -1 \implies P_i = A_i$. Find the sum of $f(P)$ over all such $P$, modulo $998244353$. ## Constraints * $1\leq N\leq 200$ * $A_i = -1$ or $1\leq A_i\leq N$. * $A_i\neq A_j$ if $A_i\neq -1$ and $A_j\neq -1$, for $i\neq j$. ## Input Input is given from Standard Input in the following format: $N$ $A_1$ $A_2$ $\ldots$ $A_N$ [samples]

Samples

Input #1

4
1 -1 4 -1

Output #1

41

We want to find the sum of $f(P)$ over two permutations $P = (1,2,4,3)$ and $P = (1,3,4,2)$. The figure below shows the impassable squares for each of them, where `.` and `#` represent a passable and impassable square, respectively.

  ......         ......
  .#....         .#....
  ..#...         ...#..
  ....#.         ....#.
  ...#..         ..#...
  ......         ......
P=(1,2,4,3)    P=(1,3,4,2)

*   For $P = (1,2,4,3)$, we have $f(P) = 18$.
*   For $P = (1,3,4,2)$, we have $f(P) = 23$.

The answer is the sum of them: $41$.

Input #2

4
4 3 2 1

Output #2

2

In this sample, we want to compute $f(P)$ for just one permutation $P = (4,3,2,1)$.

Input #3

3
-1 -1 -1

Output #3

48

*   For $P = (1,2,3)$, we have $f(P) = 10$.
*   For $P = (1,3,2)$, we have $f(P) = 6$.
*   For $P = (2,1,3)$, we have $f(P) = 6$.
*   For $P = (2,3,1)$, we have $f(P) = 12$.
*   For $P = (3,1,2)$, we have $f(P) = 12$.
*   For $P = (3,2,1)$, we have $f(P) = 2$.

The answer is the sum of them: $48$.

API Response (JSON)

{
  "problem": {
    "name": "Avoid Permutations",
    "description": {
      "content": "Let us consider the problem below for a permutation $P = (P_1, P_2, \\ldots, P_N)$ of $(1, 2, \\ldots, N)$, and denote the answer by $f(P)$. > We have an $(N+2)\\times (N+2)$ grid, where the rows are nu",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc118_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Let us consider the problem below for a permutation $P = (P_1, P_2, \\ldots, P_N)$ of $(1, 2, \\ldots, N)$, and denote the answer by $f(P)$.\n\n> We have an $(N+2)\\times (N+2)$ grid, where the rows are nu...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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