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 Kotlin (1.0.0) (judge verdict: Accepted)

fun main(args: Array<String>) {
    val mod = 1000000007L
    val fib = arrayOf<Long>(1, 2, 3, 5, 8, 13, 21, 34, 55)
    val (h, w, k) = readLine()!!.split(' ').map(String::toInt)
    val dp = Array(h + 1) { Array(w) { 0L } }
    dp[0][0] = 1
    for (i in 0 until h) {
        for (j in 0 until w) {
            if (j > 0) {
                dp[i + 1][j - 1] += dp[i][j] * fib[Math.max(0, j - 2)] * fib[Math.max(0, w - j - 2)]
                dp[i + 1][j - 1] %= mod
            }
            dp[i + 1][j] += dp[i][j] * fib[Math.max(0, j - 1)] * fib[Math.max(0, w - j - 2)]
            dp[i + 1][j] %= mod
            if (j < w - 1) {
                dp[i + 1][j + 1] += dp[i][j] * fib[Math.max(0, j - 1)] * fib[Math.max(0, w - j - 3)]
                dp[i + 1][j + 1] %= mod
            }
        }
    }
    println(dp[h][k - 1])
}