UV 6177 The King's Ups and Downs

The question is how many values in the full arrangement of 1-N are displayed in the descending or descending order.

This sort is called alternating permutations or Extremal Permutations.

You can use DP.

Dp (n, k) indicates that the length is n, the last number is k, and the last two numbers are incremental;

Dp2 (n, k) indicates that the length is n, the last number is k, and the last two numbers are in descending order;

So:

Dp (n, k) = dp2 (n, n + 1-k );

For example, 132 (low) is equivalent to 312 (high), and the relative position is equal to 4.

We will consider the last digit of dp [n] [k] as follows:

The last digit is k, because the last two digits of dp [n] [k] are incremental, so the maximum value of the nth digit is K-1. We can easily derive the DP equation:

 
 

Also:

 

So: dp (n, k) = dp (n-1, n + 1-k) + dp (n, k-1 );

The boundary condition is omitted.

Code:

[Cpp]
# Include <iostream>
# Include <stdio. h>
# Include <string. h>
# Include <stdlib. h>
Using namespace std;
 
# Define Maxn 25
# Define LL long
LL dp [Maxn] [Maxn];
LL ans [Maxn];
 
 
Void init ()
{
Memset (dp, 0, sizeof (dp ));
Memset (ans, 0, sizeof (ans ));
Dp [1] [1] = 1;
Ans [1] = 1;
For (int I = 2; I <= 20; I ++)
{
For (int k = 2; k <= I; k ++)
{
Dp [I] [k] = dp [I-1] [I + 1-k] + dp [I] [k-1];
Ans [I] + = dp [I] [k];
}
Ans [I] * = 2;
// Printf ("% lld \ n", ans [I]);
}
}
Int main ()
{
# Ifndef ONLINE_JUDGE
Freopen ("in.txt", "r", stdin );
# Endif
Init ();
Int p;
Int m, n;
Scanf ("% d", & p );
While (p --)
{
Scanf ("% d", & m, & n );
Printf ("% d % lld \ n", m, ans [n]);
}
Return 0;
}