（卡特兰数）Train Problem II --HDOJ

Train Problem II
Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65536/32768 K (Java/Others)
Total Submission(s): 909 Accepted Submission(s): 542

Problem Description
As we all know the Train Problem I, the boss of the Ignatius Train Station want to know if all the trains come in strict-increasing order, how many orders that all the trains can get out of the railway.

Input
The input contains several test cases. Each test cases consists of a number N(1<=N<=100). The input is terminated by the end of file.

Output

        For each test case, you should output how many ways that all the trains can get out of the railway.

Sample Input
1
2
3
10

Sample Output
1
2
5
16796

Hint
The result will be very large, so you may not process it by 32-bit integers.

Author
Ignatius.L

总结：
卡特兰属计算公式：
（2N！）/（（N+1）* N！*N!）
也就是
C(2N，N) / （N+1）

卡特兰数几种常用的情形：
1、括号配对问题：依照乘法结合律，给一个式子添加括号

2、合法的出栈顺序

3、将凸多边形划分成三角形

4、给N个顶点，求不同的二叉树的数量

import java.math.BigInteger;import java.util.Scanner;public class Main {    public static void main(String[] args)     {        BigInteger sum = BigInteger.ZERO;        BigInteger []f = new BigInteger[101];        f[0] = BigInteger.ONE;        for(int i =1;i<=100;i++){            f[i] = f[i-1].multiply(BigInteger.valueOf((4*i-2))).divide(BigInteger.valueOf(i+1)).multiply(BigInteger.valueOf(i));        }        Scanner cin = new Scanner(System.in);        while(cin.hasNext())        {            int n = cin.nextInt();            BigInteger ans = BigInteger.valueOf(1);            BigInteger t = BigInteger.valueOf(1);            for(int i=n+1; i<=2*n; ++i)                ans = ans.multiply(BigInteger.valueOf(i));            for(int i=2; i<=n; ++i)                t = t.multiply(BigInteger.valueOf(i));            t = t.multiply(BigInteger.valueOf(n+1));            BigInteger res = ans.divide(t);            System.out.println(res);        }    }}