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"
	"strconv"
	"strings"
)

const mod int = 1e9 + 7

func main() {
	var h, w, k int
	fmt.Scan(&h, &w, &k)

	dp := [101][8]int{}
	dp[0][0] = 1

	for i := 0; i < h; i++ {
		for bits := 0; bits < 1<<(w-1); bits++ {
			ls := fmt.Sprintf("%0"+strconv.Itoa(w-1)+"b", bits)
			if strings.Contains(ls, "11") {
				continue
			}
			for j := 0; j < w; j++ {
				if 1 <= j && ls[j-1] == '1' {
					dp[i+1][j-1] += dp[i][j]
					dp[i+1][j-1] %= mod
				} else if j < len(ls) && ls[j] == '1' {
					dp[i+1][j+1] += dp[i][j]
					dp[i+1][j+1] %= mod
				} else {
					dp[i+1][j] += dp[i][j]
					dp[i+1][j] %= mod
				}
			}
		}
	}

	fmt.Println(dp[h][k-1])
}