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: Wrong Answer)

package main

import (
	"fmt"
)

func calc(h int, w int, k int) int {
	if w == 1 {
		return 1
	}
	ei := w
	for i := w - 3; 0 < i; i-- {
		ei += i
	}
	arr := make([][]int, h)
	for i := 0; i < h; i++ {
		arr[i] = make([]int, w)
	}
	arr[0][0] = w - 1
	arr[0][1] = 1
	left := ei * ei
	for i := 1; i < h; i++ {
		sum := 0
		for j := w - 1; 0 < j; j-- {
			arr[i][j] = (arr[i-1][j] + arr[i-1][j-1])
			sum += arr[i][j]
		}
		arr[i][0] = (left - sum)
		left *= ei
	}
	return arr[h-1][k-1]
}

func main() {
	// mod := 1000000007
	var h, w, k int
	fmt.Scan(&h)
	fmt.Scan(&w)
	fmt.Scan(&k)
	fmt.Println(calc(h, w, k))
}