Chmax

AtCoder

IDarc171_b

Time2000ms

Memory256MB

Difficulty—

For a permutation $P = (P_1, P_2, \dots, P_N)$ of $(1, 2, \dots, N)$, we define $F(P)$ by the following procedure: * There is a sequence $B = (1, 2, \dots, N)$. As long as there is an integer $i$ such that $B_i \lt P_{B_i}$, perform the following operation: * Let $j$ be the smallest integer $i$ that satisfies $B_i \lt P_{B_i}$. Then, replace $B_j$ with $P_{B_j}$. Define $F(P)$ as the $B$ at the end of this process. (It can be proved that the process terminates after a finite number of steps.) You are given a sequence $A = (A_1, A_2, \dots, A_N)$ of length $N$. How many permutations $P$ of $(1,2,\dots,N)$ satisfy $F(P) = A$? Find the count modulo $998244353$. ## Constraints * $1 \leq N \leq 2 \times 10^5$ * $1 \leq A_i \leq N$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $A_1$ $A_2$ $\dots$ $A_N$ [samples]

Samples

Input #1

4
3 3 3 4

Output #1

1

For example, if $P = (2, 3, 1, 4)$, then $F(P)$ is determined to be $(3, 3, 3, 4)$ by the following steps:

*   Initially, $B = (1, 2, 3, 4)$.
*   The smallest integer $i$ such that $B_i \lt P_{B_i}$ is $1$. Replace $B_1$ with $P_{B_1} = 2$, making $B = (2, 2, 3, 4)$.
*   The smallest integer $i$ such that $B_i \lt P_{B_i}$ is $1$. Replace $B_1$ with $P_{B_1} = 3$, making $B = (3, 2, 3, 4)$.
*   The smallest integer $i$ such that $B_i \lt P_{B_i}$ is $2$. Replace $B_2$ with $P_{B_2} = 3$, making $B = (3, 3, 3, 4)$.
*   There are no more $i$ that satisfy $B_i \lt P_{B_i}$, so the process ends. The current $B = (3, 3, 3, 4)$ is defined as $F(P)$.

There is only one permutation $P$ such that $F(P) = A$, which is $(2, 3, 1, 4)$.

Input #2

4
2 2 4 3

Output #2

Input #3

8
6 6 8 4 5 6 8 8

Output #3

API Response (JSON)

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    "name": "Chmax",
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      "content": "For a permutation $P = (P_1, P_2, \\dots, P_N)$ of $(1, 2, \\dots, N)$, we define $F(P)$ by the following procedure: *   There is a sequence $B = (1, 2, \\dots, N)$.       As long as there is an integer",
      "description_type": "Markdown"
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    "platform": "AtCoder",
    "limit": {
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      "memory_limit": 262144
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    "difficulty": "None",
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    "sync_url": null,
    "sign": "arc171_b"
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      "statement_type": "Markdown",
      "content": "For a permutation $P = (P_1, P_2, \\dots, P_N)$ of $(1, 2, \\dots, N)$, we define $F(P)$ by the following procedure:\n\n*   There is a sequence $B = (1, 2, \\dots, N)$.  \n    As long as there is an integer...",
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      "language": "English"
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