CF1326F2 Wise Men (Hard Version)

This is the hard version of the problem. The difference is constraints on the number of wise men and the time limit. You can make hacks only if all versions of this task are solved. $ n $ wise men live in a beautiful city. Some of them know each other. For each of the $ n! $ possible permutations $ p_1, p_2, \ldots, p_n $ of the wise men, let's generate a binary string of length $ n-1 $ : for each $ 1 \leq i < n $ set $ s_i=1 $ if $ p_i $ and $ p_{i+1} $ know each other, and $ s_i=0 $ otherwise. For all possible $ 2^{n-1} $ binary strings, find the number of permutations that produce this binary string.

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In the first test, each wise man knows each other, so every permutation will produce the string $ 11 $ . In the second test: - If $ p = \{1, 2, 3, 4\} $ , the produced string is $ 101 $ , because wise men $ 1 $ and $ 2 $ know each other, $ 2 $ and $ 3 $ don't know each other, and $ 3 $ and $ 4 $ know each other; - If $ p = \{4, 1, 2, 3\} $ , the produced string is $ 110 $ , because wise men $ 1 $ and $ 4 $ know each other, $ 1 $ and $ 2 $ know each other and $ 2 $ , and $ 3 $ don't know each other; - If $ p = \{1, 3, 2, 4\} $ , the produced string is $ 000 $ , because wise men $ 1 $ and $ 3 $ don't know each other, $ 3 $ and $ 2 $ don't know each other, and $ 2 $ and $ 4 $ don't know each other.