Tomori has N friends. Some of the friends actually remembered her birthday!i, he or she will remember Tomori's birthday at time ti and give her a birthday wish at that instant (or not remember her birthday at all if. Some of Tomori's friends are very considerate however, and decides to remind others about her birthday. Friend i will remind friend pi about Tomori's birthday after he or she has wished her. Formally, if Friend i gave Tomori a birthday wish at time t, then he or she will remind friend pi to give Tomori a birthday wish at time t+1 (and friend pi will give her a birthday wish at time t+1 and remind friend. =i is possible, which means that the friend will not remind anyone about Tomori's birthday after sending their wish.The day is over and at the end of the day, Tomori receives the wish from Friend i at time. }, how many possible different days she could have experienced. Two permutations p, q correspond to different days if at some point of time, the number of friends who wished Tomori was different.
Today is Tomori's birthday! Usually, people get many birthday wishes on social media, but Tomori has decided to hide her birthday on social media, and thus does not expect many (if any) wishes from her friends. Tomori has N friends. Some of the friends actually remembered her birthday! For each friend i i, he or she will remember Tomori's birthday at time ti and give her a birthday wish at that instant (or not remember her birthday at all if t_i = -1 t i =−1). Some of Tomori's friends are very considerate however, and decides to remind others about her birthday. Friend i will remind friend pi about Tomori's birthday after he or she has wished her. Formally, if Friend i gave Tomori a birthday wish at time t, then he or she will remind friend pi to give Tomori a birthday wish at time t+1 (and friend pi will give her a birthday wish at time t+1 and remind friend p_{p_{i}} p p i at time t+2 and so on, assuming that they have not already gave her a birthday wish). The sequence pi forms a permutation, i.e. each friend will be reminded by exactly one friend. Note that p_i = i p i =i is possible, which means that the friend will not remind anyone about Tomori's birthday after sending their wish. The day is over and at the end of the day, Tomori receives the wish from Friend i at time b_i b i (if b_i = -1 b i =−1, it means that friend i never wished her for her birthday). Note that b_i b i might not be equal to t_i t i as he or she might be reminded by another friend at an earlier time. Tomori receives the wishes, but she actually can't distinguish between her friends, so she only knows the number of wishes she received at each particular time. Now, she wonders, over all possible permutations {p_i} {p i }, how many possible different days she could have experienced. Two permutations p, q correspond to different days if at some point of time, the number of friends who wished Tomori was different.
