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"
)

func main() {
	var h, w, k int64
	const m = 1000*1000*1000 + 7
	fmt.Scan(&h, &w, &k)

	if w == 1 {
		if k == 1 {
			fmt.Println(1)
		} else {
			fmt.Println(0)
		}
		return
	}
	var z [9][2]int64
	z[0][0] = 1
	z[0][1] = 1
	z[1][0] = 1
	z[1][1] = 1
	for i := range iter(2, 9) {
		z[i][0] = z[i-1][0] + z[i-1][1]
		z[i][1] = z[i-1][0]
	}

	var dp [101][8]int64
	dp[0][0] = 1
	for i := range iter(1, h+1) {
		for j := range iter(0, w) {
			l := z[j][0]
			r := z[w-j-1][0]
			dp[i][j] = (dp[i-1][j] * (l * r)) % m
		}
		for j := range iter(0, w-1) {
			l := z[j][0]
			r := z[w-j-2][0]
			dp[i][j] = (dp[i][j] + (l*r)*dp[i-1][j+1]) % m
			dp[i][j+1] = (dp[i][j+1] + (l*r)*dp[i-1][j]) % m
		}
	}
	fmt.Println(dp[h][k-1])
}

func iter(l int64, r int64) chan int64 {
	var i int64 = l
	c := make(chan int64)
	run := func() {
		for i < r {
			c <- i
			i += 1
		}
		close(c)
	}
	go run()
	return c
}