Special Subsets

AtCoder

IDarc114_b

Time2000ms

Memory256MB

Difficulty—

Let $S$ be a set composed of all integers from $1$ through $N$. $f$ is a function from $S$ to $S$. You are given the values $f(1), f(2), \cdots, f(N)$ as $f_1, f_2, \cdots, f_N$. Find the number, modulo $998244353$, of non-empty subsets $T$ of $S$ satisfying both of the following conditions: * For every $a \in T$, $f(a) \in T$. * For every $a, b \in T$, $f(a) \neq f(b)$ if $a \neq b$. ## Constraints * $1 \leq N \leq 2 \times 10^5$ * $1 \leq f_i \leq N$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $f_1$ $f_2$ $\ldots$ $f_N$ [samples]

Samples

Input #1

2
2 1

Output #1

1

We have $f(1) = 2, f(2) = 1$. Since $f(1) \neq f(2)$, the second condition is always satisfied, but the first condition requires $T$ to contain $1$ and $2$ simultaneously.

Input #2

2
1 1

Output #2

1

We have $f(1) = f(2) = 1$. The first condition requires $T$ to contain $1$, and the second condition forbids $T$ to contain $2$.

Input #3

3
1 2 3

Output #3

7

We have $f(1) = 1, f(2) = 2, f(3) = 3$. Both of the conditions are always satisfied, so all non-empty subsets of $T$ count.

API Response (JSON)

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    "name": "Special Subsets",
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      "content": "Let $S$ be a set composed of all integers from $1$ through $N$. $f$ is a function from $S$ to $S$. You are given the values $f(1), f(2), \\cdots, f(N)$ as $f_1, f_2, \\cdots, f_N$. Find the number, modu",
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    {
      "statement_type": "Markdown",
      "content": "Let $S$ be a set composed of all integers from $1$ through $N$.\n$f$ is a function from $S$ to $S$. You are given the values $f(1), f(2), \\cdots, f(N)$ as $f_1, f_2, \\cdots, f_N$.\nFind the number, modu...",
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