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 (h, w, k) = readLine()!!.split(" ").map { it.toInt() }
    if(w==1){
        println(1)
        return
    }

    val patterns = (0 until 1.shl(w-1)).filter{ bit -> (0..8).all{ (bit shr it) and 3 != 3 } }
    val rowPatterns=IntArray(w-1)
    patterns.forEach {
        for (i in 0 until w-1){
            rowPatterns[i]+= it shr i and 1
        }
    }

    val dp=LongArray(w)
    dp[0]=1

    repeat(h){
        val nextDp=LongArray(w)
        nextDp[0] = dp[0]*(patterns.size-rowPatterns[0]) + dp[1]*rowPatterns[0]
        nextDp[w-1] = dp[w-1]*(patterns.size-rowPatterns[w-2]) + dp[w-2]*rowPatterns[w-2]
        for(i in 1..w-2){
            nextDp[i] = dp[i]*(patterns.size-rowPatterns[i-1]-rowPatterns[i]) +
                        dp[i-1]*rowPatterns[i-1] +
                        dp[i+1]*rowPatterns[i]
        }
        for(i in 0..w-1){
            dp[i] = nextDp[i] % 1000000007L
        }
    }

    println(dp[k-1])
 }