CF1936E Yet Yet Another Permutation Problem

You are given a permutation $ p $ of length $ n $ . Please count the number of permutations $ q $ of length $ n $ which satisfy the following: - for each $ 1 \le i < n $ , $ \max(q_1,\ldots,q_i) \neq \max(p_1,\ldots,p_i) $ . Since the answer may be large, output the answer modulo $ 998\,244\,353 $ .

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 10^4 $ ). The description of the test cases follows. The first line of each test case contains a single integer $ n $ ( $ 2 \le n \le 2 \cdot 10^5 $ ). The second line of each test case contains $ n $ integers $ p_1, p_2, \ldots, p_n $ ( $ 1 \le p_i \le n $ ). It is guaranteed that $ p $ is a permutation. It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 2 \cdot 10^5 $ .

For each test case, print a single integer — the answer modulo $ 998\,244\,353 $ .

In the first test case, $ p = [2, 1] $ . The only suitable $ q $ is $ [1, 2] $ . Indeed, we need to satisfy the inequality $ q_1 \neq p_1 $ . It only holds for $ q = [1, 2] $ . In the second test case, $ p = [1, 2, 3] $ . So $ q $ has to satisfy two inequalities: $ q_1 \neq p_1 $ and $ \max(q_1, q_2) \neq \max(1, 2) = 2 $ . One can prove that this only holds for the following $ 3 $ permutations: - $ q = [2, 3, 1] $ : in this case $ q_1 = 2 \neq 1 $ and $ \max(q_1, q_2) = 3 \neq 2 $ ; - $ q = [3, 1, 2] $ : in this case $ q_1 = 3 \neq 1 $ and $ \max(q_1, q_2) = 3 \neq 2 $ ; - $ q = [3, 2, 1] $ : in this case $ q_1 = 3 \neq 1 $ and $ \max(q_1, q_2) = 3 \neq 2 $ .