Problem:
Score: 400 points

Problem Statement

Amidakuji is a traditional method of lottery in Japan.

To make an amidakuji, we first draw W parallel vertical lines, and then draw horizontal lines that connect them. The length of each vertical line is H+1 [cm], and the endpoints of the horizontal lines must be at 1, 2, 3, ..., or H [cm] from the top of a vertical line.

A valid amidakuji is an amidakuji that satisfies the following conditions:

No two horizontal lines share an endpoint.

The two endpoints of each horizontal lines must be at the same height.

A horizontal line must connect adjacent vertical lines.

Find the number of the valid amidakuji that satisfy the following condition, modulo 1\ 000\ 000\ 007: if we trace the path from the top of the leftmost vertical line to the bottom, always following horizontal lines when we encounter them, we reach the bottom of the K-th vertical line from the left.

For example, in the following amidakuji, we will reach the bottom of the fourth vertical line from the left.

Constraints

H is an integer between 1 and 100 (inclusive).

W is an integer between 1 and 8 (inclusive).

K is an integer between 1 and W (inclusive).

Input

Input is given from Standard Input in the following format:

H W K

Output

Print the number of the amidakuji that satisfy the condition, modulo 1\ 000\ 000\ 007.

Sample Input 1

1 3 2

Sample Output 1

1

Only the following one amidakuji satisfies the condition:

Sample Input 2

1 3 1

Sample Output 2

2

Only the following two amidakuji satisfy the condition:

Sample Input 3

2 3 3

Sample Output 3

1

Only the following one amidakuji satisfies the condition:

Sample Input 4

2 3 1

Sample Output 4

5

Only the following five amidakuji satisfy the condition:

Sample Input 5

7 1 1

Sample Output 5

1

As there is only one vertical line, we cannot draw any horizontal lines. Thus, there is only one amidakuji that satisfies the condition: the amidakuji with no horizontal lines.

Sample Input 6

15 8 5

Sample Output 6

437760187

Be sure to print the answer modulo 1\ 000\ 000\ 007.

Solution in Go (1.6) (judge verdict: Accepted)

package main

import (
	"fmt"
)

const mod = 1000000007

func main() {
	var H, W, K int
	fmt.Scanf("%d%d%d", &H, &W, &K)
	dp := make([][]int, H+1)
	for i := 0; i <= H; i++ {
		dp[i] = make([]int, W+1)
	}
	cp := make([]int, W+1)
	cp[0] = 1
	cp[1] = 1
	for i := 2; i <= W; i++ {
		cp[i] = cp[i-1] + cp[i-2]
	}
	dp[0][1] = 1
	for i := 0; i < H; i++ {
		for j := 1; j <= W; j++ {
			if j >= 2 {
				dp[i+1][j-1] += dp[i][j] * cp[j-2] % mod * cp[W-j] % mod
				dp[i+1][j-1] %= mod
			}
			dp[i+1][j] += dp[i][j] * cp[j-1] % mod * cp[W-j] % mod
			dp[i+1][j] %= mod
			if j <= W-1 {
				dp[i+1][j+1] += dp[i][j] * cp[j-1] % mod * cp[W-j-1] % mod
			}
		}
	}
	fmt.Printf("%d\n", dp[H][K])
	return
}