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
ab_hi_shek111:
20170912 23:07:26
solved and Motivated :) 

aditya9125:
20170911 22:33:34
Try iterative DP, you will definitely enjoy. 

ashishranjan28:
20170523 08:43:04
easy recursion :) 

sultania23:
20170324 08:10:46
recursion with mem... nice one.. 

ayush_1997:
20170221 19:06:27
nice problem :) 

Mayank Agarwal:
20170129 20:39:28
nice dp:) 

sushantoberoi3:
20170115 06:41:49
easy problem.. 

prasoonbatham:
20170114 13:25:35
good problem :) 

vaibhav goyal:
20160821 17:17:43
Nice DP problem .. .enjoyed solving it...some good thinking is required :D 

nonushikhar:
20160627 20:12:45
nice problem :) 
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 