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 provides
//
// File:  d.go
// Author: ymiyamoto
//
// Created on Mon Dec 17 18:43:46 2018
//
package main

import "fmt"

const MOD = 1e9 + 7

func adjacent(net int) bool {
	for i := 0; i < 7; i++ {
		if net&(0x03<<uint8(i)) == 0x03<<uint8(i) {
			return true
		}
	}
	return false
}

func main() {
	var H, W, K int
	fmt.Scan(&H, &W, &K)
	dp := make([][]int, H+1)
	for i := range dp {
		dp[i] = make([]int, W)
	}
	dp[0][0] = 1

	for h := 1; h < H+1; h++ {
		for w := 0; w < len(dp[h]); w++ {
			for i := 0; i < 1<<uint8(W-1); i++ {
				if adjacent(i) {
					continue
				}
				if w+1 < W && i&(1<<uint8(w)) != 0 {
					dp[h][w] += dp[h-1][w+1] % MOD
				} else if w > 0 && i&(1<<uint8(w-1)) != 0 {
					dp[h][w] += dp[h-1][w-1] % MOD
				} else {
					dp[h][w] += dp[h-1][w] % MOD
				}
				dp[h][w] %= MOD
			}
		}
	}

	fmt.Println(dp[H][K-1])
}