N. Count Permutations
math
You are given a string \(s\) with length \(n-1\) whose characters are either \(\texttt{<}\) or \(\texttt{>}\).Count the permutations \(p_1, \, p_2, \, \dots, \, p_n\) of \(1, \, 2, \, \dots, \, n\) such that, for all \(i = 1, \, 2, \, \dots, \, n - 1\), if \(s_i\) is \(\texttt{<}\) then \(p_i < p_{i+1}\) and if \(s_i\) is \(\texttt{>}\) then \(p_i > p_{i+1}\).Since this number can be very large, compute its logarithm in base \(2\).
The first line contains a single integer \(n\) (\(2 \le n \le 100\,000\)).The second line contains a string \(s\) of length \(n-1\); each character of \(s\) is either \(\texttt{<}\) or \(\texttt{>}\).
Print the logarithm in base \(2\) of the number of permutations satisfying the constraints described in the statement.Your answer is considered correct if its absolute or relative error does not exceed \(10^{-6}\). Formally, let your answer be \(x\) and let the correct answer be \(y\). Your answer is accepted if and only if \(\frac{|x - y|}{\max{(1, |y|)}} \le 10^{-6}\).
In the first sample, there is only one valid permutation, that is \([2, 1]\). Since \(\log_2(1)=0\), the correct output is \(0\).In the second sample, there are \(2\) valid permutations, that are \([3, 1, 2]\) and \([2, 1, 3]\). Since \(\log_2(2)=1\), the correct output is \(1\).In the third sample, there are \(4\) valid permutations, that are \([1, 5, 4, 3, 2]\), \([2, 5, 4, 3, 1]\), \([3, 5, 4, 2, 1]\), \([4, 5, 3, 2, 1]\). Since \(\log_2(4)=2\), the correct output is \(2\).In the fourth sample, there are \(909\) valid permutations. Notice that \(\log_2(909)=9.828136484194\ldots\)