SQRBR - Square Brackets


You are given:

  • a positive integer n,
  • an integer k, 1<=k<=n,
  • an increasing sequence of k integers 0 < s1 < s2 < ... < sk <= 2n.

What is the number of proper bracket expressions of length 2n with opening brackets appearing in positions s1, s2,...,sk?

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 s1,s2,...,sk,
  • 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 i-th line of output should contain one integer - the number of proper bracket expressions of length 2n with opening brackets appearing at positions s1, s2,...,sk.

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: 2017-09-12 23:07:26

solved and Motivated :)

aditya9125: 2017-09-11 22:33:34

Try iterative DP, you will definitely enjoy.

ashishranjan28: 2017-05-23 08:43:04

easy recursion :)

sultania23: 2017-03-24 08:10:46

recursion with mem... nice one..

ayush_1997: 2017-02-21 19:06:27

nice problem :)

Mayank Agarwal: 2017-01-29 20:39:28

nice dp:)

sushantoberoi3: 2017-01-15 06:41:49

easy problem..

prasoonbatham: 2017-01-14 13:25:35

good problem :)

vaibhav goyal: 2016-08-21 17:17:43

Nice DP problem .. .enjoyed solving it...some good thinking is required :D

nonushikhar: 2016-06-27 20:12:45

nice problem :)


Added by:adrian
Date:2004-06-22
Time limit:3s
Source limit:50000B
Memory limit:1536MB
Cluster: Cube (Intel G860)
Languages:All
Resource:III Polish Collegiate Team Programming Contest (AMPPZ), 1998