CF1613F Tree Coloring

You are given a rooted tree consisting of $ n $ vertices numbered from $ 1 $ to $ n $ . The root of the tree is the vertex $ 1 $ . You have to color all vertices of the tree into $ n $ colors (also numbered from $ 1 $ to $ n $ ) so that there is exactly one vertex for each color. Let $ c_i $ be the color of vertex $ i $ , and $ p_i $ be the parent of vertex $ i $ in the rooted tree. The coloring is considered beautiful if there is no vertex $ k $ ( $ k > 1 $ ) such that $ c_k = c_{p_k} - 1 $ , i. e. no vertex such that its color is less than the color of its parent by exactly $ 1 $ . Calculate the number of beautiful colorings, and print it modulo $ 998244353 $ .

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