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 

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

hide comments
2017-01-14 13:25:35
good problem :)
2016-08-21 17:17:43 vaibhav goyal
Nice DP problem .. .enjoyed solving it...some good thinking is required :D
2016-06-27 20:12:45
nice problem :)
2016-05-29 08:45:24 Nguyen Cuong
The idea is fairly similar to MREPLBRC if you've done MREPLBRC first :)
2016-01-24 12:59:08 Sumit Vohra
Aur Bhai Savil kaisa hai


Last edit: 2016-02-15 19:46:34
2015-12-13 20:14:47 Mit
good question. do try MPILOT after this as someone here already mentioned.
2015-11-10 08:45:48 :.Mohib.:
A very nice question....!!
2015-10-14 10:00:18 Anoop Narang
how come answer for last 2nd case is 3. I can only count 2 ways.
2015-10-07 20:17:01 priyank
nice dp :) My 150th
2015-09-12 21:04:45
Nice Problem !
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