{"problem":{"name":"Special Subsets","description":{"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","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"arc114_b"},"statements":[{"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, modulo $998244353$, of non-empty subsets $T$ of $S$ satisfying both of the following conditions:\n\n*   For every $a \\in T$, $f(a) \\in T$.\n    \n*   For every $a, b \\in T$, $f(a) \\neq f(b)$ if $a \\neq b$.\n\n## Constraints\n\n*   $1 \\leq N \\leq 2 \\times 10^5$\n*   $1 \\leq f_i \\leq N$\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$f_1$ $f_2$ $\\ldots$ $f_N$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"arc114_b","tags":[],"sample_group":[["2\n2 1","1\n\nWe 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."],["2\n1 1","1\n\nWe have $f(1) = f(2) = 1$. The first condition requires $T$ to contain $1$, and the second condition forbids $T$ to contain $2$."],["3\n1 2 3","7\n\nWe 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."]],"created_at":"2026-03-03 11:01:14"}}