{"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 operation until all the colors of the balls become the same.\n\n*   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 $P_i$.\n\nFind the expected value of number of operations, modulo $998244353$.\nHere 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.\n\n## Constraints\n\n*   $2 \\le N \\le 2000$\n*   $1 \\le A_i \\le N$\n*   All values in input are integers.\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$\n$A_1$ $A_2$ $\\dots$ $A_N$\n\n[samples]\n\n## Notes\n\nWe 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 $R$.","is_translate":false,"language":"English"}],"meta":{"iden":"abc249_h","tags":[],"sample_group":[["2\n1 2","4\n\nThe 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 $4$."],["3\n1 1 1","0\n\nThe operation is never performed, since the colors of the balls are already the same."],["10\n3 1 4 1 5 9 2 6 5 3","900221128"]],"created_at":"2026-03-03 11:01:13"}}