The King’s Ups and Downs
The king has guards of all different heights. Rather than line them up in increasing or decreasing height order, he wants to line them up so each guard is either shorter than the guards next to him or taller than the guards next to him (so the heights go up and down along the line). For example, seven guards of heights 160, 162, 164, 166, 168, 170 and 172 cm. could be arranged as:
or perhaps:
The king wants to know how many guards he needs so he can have a different up and down order at each changing of the guard for rest of his reign. To be able to do this, he needs to know for a given number of guards, n, how many different up and down orders there are:
For example, if there are four guards: 1, 2, 3,4 can be arrange as:
1324, 2143, 3142, 2314, 3412, 4231, 4132, 2413, 3241, 1423
For this problem, you will write a program that takes as input a positive integer n, the number of guards and returns the number of up and down orders for n guards of differing heights.
Input
The first line of input contains a single integer P, (1 <= P <= 1000), which is the number of data sets that follow. Each data set consists of single line of input containing two integers. The first integer, D is the data set number. The second integer, n (1 <= n <= 20), is the number of guards of differing heights.
Output
For each data set there is one line of output. It contains the data set number (D) followed by a single space, followed by the number of up and down orders for the n guards.
Sample Input
4
1 1
2 3
3 4
4 20
Sample Output
1 1
2 4
3 10
4 740742376475050
题目大意：给你一组不同身高的士兵，让你按高低高低高低或低高低高低高的方式排成一排，并输出这组的排列方式的数目。
解题思想：
当你增加一个数n并按题目方式加入前边n-1个数之间时，你可以有n个位置插入
1 2 3 4 5 6 7 加入n=8
0 1 2 3 4 5 6 7 可插8八个位置
当我们插入n时必须要满足它的前边高->低 - n - 低->高 因为加入的n是最大的。
如果我们我们取第i个位置插入那么前i位置以前共有i个数，那么这i个数将有C[n-1][i],即从前n-1个数中取i个数的可能方式数（排列组合）。
记录前边i个数中为高低排列的为dp[i][0],低高排列的为[i][1]，由对称性可知两个是相等的且等于sum[i]/2.
那么可以得到公式
AC代码：
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