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 (1.14.1) (judge verdict: Accepted)

package main

import (
	"fmt"
	"math"
)

func main() {
	MOD := int(math.Pow10(9) + 7)
	var H, W, K int
	fmt.Scan(&H, &W, &K)

	DP := make([][]int, H+1)
	for i := 0; i <= H; i++ {
		DP[i] = make([]int, W)
	}
	DP[0][0] = 1

	for i := 1; i <= H; i++ {
		// jはW-1桁(0~W-2桁目まで)
		for j := 0; j < (1 << (W - 1)); j++ {
			flg := true
			for k := 0; k < (W - 2); k++ {
				if (j&(1<<k))*(j&(1<<(k+1))) > 0 {
					flg = false
					break
				}
			}

			if !flg {
				continue
			}

			for k := 0; k < W; k++ {
				if k == 0 {
					if j&(1<<k) > 0 {
						DP[i][k] += DP[i-1][k+1]
					} else {
						DP[i][k] += DP[i-1][k]
					}
				} else if k == W-1 {
					if j&(1<<(k-1)) > 0 {
						DP[i][k] += DP[i-1][k-1]
					} else {
						DP[i][k] += DP[i-1][k]
					}
				} else {
					if j&(1<<k) > 0 {
						DP[i][k] += DP[i-1][k+1]
					} else if j&(1<<(k-1)) > 0 {
						DP[i][k] += DP[i-1][k-1]
					} else {
						DP[i][k] += DP[i-1][k]
					}
				}
				DP[i][k] %= MOD
			}
		}
	}
	fmt.Println(DP[H][K-1])
}