One to One

AtCoder

IDarc140_d

Time2000ms

Memory256MB

Difficulty—

For an integer sequence $X=(X_1,X_2,\dots,X_N)$ of length $N$ whose elements are all between $1$ and $N$ (inclusive), consider the question below, and let $f(X)$ be the answer. > There is an undirected graph $G$ with $N$ vertices (and possibly multi-edges and self-loops). $G$ has $N$ edges, the $i$\-th of which connects Vertex $i$ and Vertex $X_i$. Find the number of connected components in $G$. You are given an integer sequence $A=(A_1,A_2,\dots,A_N)$ of length $N$, where each $A_i$ is an integer between $1$ and $N$ (inclusive) or $-1$. Consider an integer sequence $X=(X_1,X_2,\dots,X_N)$ of length $N$ whose elements are all between $1$ and $N$ such that $A_i \neq -1 \Rightarrow A_i = X_i$. Find the sum of $f(X)$ over all such $X$, modulo $998244353$. ## Constraints * $1 \le N \le 2000$ * $A_i$ is between $1$ and $N$ (inclusive) or $-1$. * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $A_1$ $A_2$ $\dots$ $A_N$ [samples]

Samples

Input #1

3
-1 1 3

Output #1

5

There are three sequences $X$ satisfying the requirement, as follows.

*   $X=(1,1,3)$, for which the answer to the question is $2$.
*   $X=(2,1,3)$, for which the answer to the question is $2$.
*   $X=(3,1,3)$, for which the answer to the question is $1$.

Thus, the answer is $2+2+1=5$.

Input #2

1
1

Output #2

Input #3

8
-1 3 -1 -1 8 -1 -1 -1

Output #3

API Response (JSON)

{
  "problem": {
    "name": "One to One",
    "description": {
      "content": "For an integer sequence $X=(X_1,X_2,\\dots,X_N)$ of length $N$ whose elements are all between $1$ and $N$ (inclusive), consider the question below, and let $f(X)$ be the answer. > There is an undirect",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc140_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "For an integer sequence $X=(X_1,X_2,\\dots,X_N)$ of length $N$ whose elements are all between $1$ and $N$ (inclusive), consider the question below, and let $f(X)$ be the answer.\n\n> There is an undirect...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments