CF1553I Stairs

For a permutation $ p $ of numbers $ 1 $ through $ n $ , we define a stair array $ a $ as follows: $ a_i $ is length of the longest segment of permutation which contains position $ i $ and is made of consecutive values in sorted order: $ [x, x+1, \ldots, y-1, y] $ or $ [y, y-1, \ldots, x+1, x] $ for some $ x \leq y $ . For example, for permutation $ p = [4, 1, 2, 3, 7, 6, 5] $ we have $ a = [1, 3, 3, 3, 3, 3, 3] $ . You are given the stair array $ a $ . Your task is to calculate the number of permutations which have stair array equal to $ a $ . Since the number can be big, compute it modulo $ 998\,244\,353 $ . Note that this number can be equal to zero.

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