Ex - Dye Color

AtCoder

IDabc249_h

Time3500ms

Memory256MB

Difficulty—

There are $N$ balls numbered $1$ through $N$. Initially, Ball $i$ is painted in Color $A_i$. Colors are represented by integers between $1$ and $N$, inclusive. Consider repeating the following operation until all the colors of the balls become the same. * There are $2^N$ subsets (including the empty set) of the set consisting of the $N$ balls; choose one of the subsets uniformly at random. Let $X_1,X_2,...,X_K$ be the indices of the chosen balls. Next, choose a permutation obtained by choosing $K$ integers out of integers $(1,2,\dots,N)$ uniformly at random. Let $P=(P_1,P_2,\dots,P_K)$ be the chosen permutation. For each integer $i$ such that $1 \le i \le K$, change the color of Ball $X_i$ to $P_i$. Find the expected value of number of operations, modulo $998244353$. Here a permutation obtained by choosing $K$ integers out of integers $(1,2,\dots,N)$ is a sequence of $K$ integers between $1$ and $N$, inclusive, whose elements are pairwise distinct. ## Constraints * $2 \le N \le 2000$ * $1 \le A_i \le N$ * 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] ## Notes We can prove that the sought expected value is always a rational number. Also, under the Constraints of this problem, when the value is represented by two coprime integers $P$ and $Q$ as $\frac{P}{Q}$, we can prove that there exists a unique integer $R$ such that $R \times Q \equiv P(\bmod\ 998244353)$ and $0 \le R < 998244353$. Find this $R$.

Samples

Input #1

2
1 2

Output #1

4

The operation is repeated until a subset of size $1$ is chosen and the color of the ball is changed to the color of the ball not contained in the subset. The probability is $\displaystyle \frac{2}{4} \times \frac{1}{2}=\frac{1}{4}$, so the expected value is $4$.

Input #2

3
1 1 1

Output #2

0

The operation is never performed, since the colors of the balls are already the same.

Input #3

10
3 1 4 1 5 9 2 6 5 3

Output #3

900221128

API Response (JSON)

{
  "problem": {
    "name": "Ex - Dye Color",
    "description": {
      "content": "There are $N$ balls numbered $1$ through $N$. Initially, Ball $i$ is painted in Color $A_i$. Colors are represented by integers between $1$ and $N$, inclusive. Consider repeating the following operati",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3500,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc249_h"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ balls numbered $1$ through $N$. Initially, Ball $i$ is painted in Color $A_i$.\nColors are represented by integers between $1$ and $N$, inclusive.\nConsider repeating the following operati...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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