{"problem":{"name":"Delete Range Mex","description":{"content":"You are given a positive integer $N$ and a sequence of $M$ non-negative integers $A = (A_{1}, A_{2}, \\dots, A_{M})$. Here, all elements of $A$ are distinct integers between $0$ and $N-1$, inclusive. F","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"arc178_d"},"statements":[{"statement_type":"Markdown","content":"You are given a positive integer $N$ and a sequence of $M$ non-negative integers $A = (A_{1}, A_{2}, \\dots, A_{M})$.\nHere, all elements of $A$ are distinct integers between $0$ and $N-1$, inclusive.\nFind the number, modulo $998244353$, of permutations $P$ of $(0, 1, \\dots, N-1)$ that satisfy the following condition.\n\n*   After initializing a sequence $B = (B_{1}, B_{2}, \\dots, B_{N})$ to $P$, it is possible to make $B = A$ by repeating the following operation some number of times:\n    *   Choose $l$ and $r$ such that $1 \\leq l \\leq r \\leq |B|$, and if $\\mathrm{mex}({B_{l}, B_{l+1}, \\dots, B_{r}})$ is contained in $B$, remove it from $B$.\n\nWhat is $\\mathrm{mex}(X)$? For a finite set $X$ of non-negative integers, $\\mathrm{mex}(X)$ is defined as the smallest non-negative integer that is not in $X$.\n\n## Constraints\n\n*   $1 \\leq M \\leq N \\leq 500$\n*   $0 \\leq A_{i} < N$\n*   All elements of $A$ are distinct.\n*   All input values are integers.\n\n## Input\n\nThe input is given from Standard Input in the following format:\n\n$N$ $M$\n$A_{1}$ $A_{2}$ $\\cdots$ $A_{M}$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"arc178_d","tags":[],"sample_group":[["4 2\n1 3","8\n\nAfter initializing $B = (2, 1, 0, 3)$, it is possible to make $B = A$ using the following steps:\n\n*   Choose $(l, r) = (2, 4)$, remove $\\mathrm{mex}({1, 0, 3}) = 2$ from $B$, making $B = (1, 0, 3)$.\n*   Choose $(l, r) = (3, 3)$, remove $\\mathrm{mex}({3}) = 0$ from $B$, making $B = (1, 3)$.\n\nThus, $P = (2, 1, 0, 3)$ satisfies the condition.\nThere are eight permutations $P$ that satisfy the condition, including the above, so print $8$."],["4 4\n0 3 2 1","1\n\nOnly $P = (0, 3, 2, 1)$ satisfies the condition."],["16 7\n9 2 4 0 1 6 7","3520"],["92 4\n1 67 16 7","726870122\n\nFind the count modulo $998244353$."]],"created_at":"2026-03-03 11:01:13"}}