A15585 | Count Permutations
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内存限制256MB
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题目描述
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 $2$ .
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 $2$ .
输入格式
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 $\texttt{>}$ .
The second line contains a string $s$ of length $n-1$ ; each character of $s$ is either $\texttt{<}$ or $\texttt{>}$ .
输出格式
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 $\frac{|x - y|}{\max{(1, |y|)}} \le 10^{-6}$ .
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 $\frac{|x - y|}{\max{(1, |y|)}} \le 10^{-6}$ .
输入输出样例
输入 #1
2 <
输出 #1
0.0000000000
输入 #2
3 <>
输出 #2
1.0000000000
输入 #3
5 ><<<
输出 #3
2.0000000000
输入 #4
10 <><<<<<>>
输出 #4
9.8281364842
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 $\log_2(909)=9.828136484194\ldots$
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 $\log_2(909)=9.828136484194\ldots$
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评测结果：Accepted