{"problem":{"name":"Nondivisible Prefix Sums","description":{"content":"You are given a prime number $P$, which you don't like. Let's call an array of integers $A_1, A_2, \\dots, A_N$ **good**, if it is possible to reorder the elements in such a way that no prefix sum is d","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"agc052_c"},"statements":[{"statement_type":"Markdown","content":"You are given a prime number $P$, which you don't like.\nLet's call an array of integers $A_1, A_2, \\dots, A_N$ **good**, if it is possible to reorder the elements in such a way that no prefix sum is divisible by $P$ (that is, there is **no** $i$ with $1 \\le i \\le N$ and $A_1 + A_2 + \\dots + A_i \\equiv 0 \\bmod P$ after the reordering).\nConsider all $(P-1)^N$ arrays of length $N$ with elements from $1$ to $P-1$. How many of them are **good**?\nAs this number can be very big, output it modulo $998244353$.\n\n## Constraints\n\n*   $1 \\le N \\le 5000$\n*   $2 \\le P \\le 10^8$\n*   $P$ is a prime.\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$ $P$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"agc052_c","tags":[],"sample_group":[["2 5","12\n\nThere are $12$ good arrays: $[1, 1]$, $[1, 2]$, $[1, 3]$, $[2, 1]$, $[2, 2]$, $[2, 4]$, $[3, 1]$, $[3, 3]$, $[3, 4]$, $[4, 2]$, $[4, 3]$, $[4, 4]$."],["4 3","8\n\nThere are $8$ good arrays: $[1, 1, 1, 2]$, $[1, 1, 2, 1]$, $[1, 2, 1, 1]$, $[2, 1, 1, 1]$, $[2, 2, 2, 1$\\], $[2, 2, 1, 2]$, $[2, 1, 2, 2]$, $[1, 2, 2, 2]$."],["5000 99999989","51699346"],["2021 307","644635349"]],"created_at":"2026-03-03 11:01:14"}}