SQRBR  Square Brackets
You are given:
 a positive integer n,
 an integer k, 1<=k<=n,
 an increasing sequence of k integers 0 < s_{1} < s_{2} < ... < s_{k} <= 2n.
What is the number of proper bracket expressions of length 2n with opening brackets appearing in positions s_{1}, s_{2},...,s_{k}?
Illustration
Several proper bracket expressions:
[[]][[[]][]] [[[][]]][][[]]
An improper bracket expression:
[[[][]]][]][[]]
There is exactly one proper expression of length 8 with opening brackets in positions 2, 5 and 7.
Task
Write a program which for each data set from a sequence of several data sets:
 reads integers n, k and an increasing sequence of k integers from input,
 computes the number of proper bracket expressions of length 2n with opening brackets appearing at positions s_{1},s_{2},...,s_{k},
 writes the result to output.
Input
The first line of the input file contains one integer d, 1 <= d <= 10, which is the number of data sets. The data sets follow. Each data set occupies two lines of the input file. The first line contains two integers n and k separated by single space, 1 <= n <= 19, 1 <= k <= n. The second line contains an increasing sequence of k integers from the interval [1;2n] separated by single spaces.
Output
The ith line of output should contain one integer  the number of proper bracket expressions of length 2n with opening brackets appearing at positions s_{1}, s_{2},...,s_{k}.
Example
Sample input: 5 1 1 1 1 1 2 2 1 1 3 1 2 4 2 5 7 Sample output: 1 0 2 3 2
hide comments
Vipul Pandey:
20140202 22:37:21
easy 2d dp. 

Aradhya Makkar:
20140118 14:14:27
@ xqj , maybe easy for chinese but not for the rest of us. 

siddhant bhatt:
20130913 16:09:32
dont give up! 

npsabari:
20120719 13:18:29
Easy one! though not beginners DP..


Romal Thoppilan:
20120613 16:04:52
ya i also thought it was easy enough!


Rahul Bansal:
20120607 04:36:30
beginners DP problem.


xqj:
20120410 00:05:27
It is a easy dynamic programming. Last edit: 20120410 00:22:24 

Hussain Kara Fallah:
20120403 18:24:04
Nice DP Problem 

pfiesteria:
20110917 08:17:05
It's hard for me, but finally I got AC. 

Rajesh V:
20090612 07:57:12
Hint: An exercise for dynamic programming 
Added by:  adrian 
Date:  20040622 
Time limit:  3s 
Source limit:  50000B 
Memory limit:  1536MB 
Cluster:  Cube (Intel G860) 
Languages:  All 
Resource:  III Polish Collegiate Team Programming Contest (AMPPZ), 1998 