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 = (1e9).toLong() + 7
    val (H, W, K) = readLine()!!.split(" ").map {it.toInt() }

    val dp1 = Array(W + 1) { ModInt(0, MOD) }

    dp1[0] = ModInt(1, MOD)

    for (i in 1..W) {
        dp1[i] += dp1[i - 1]
        if (i > 1) {
            dp1[i] += dp1[i - 2]
        }
    }

    val dp = Array(H + 1) { Array(W) { ModInt(0, MOD)} }

    dp[0][0] = ModInt(1, MOD)

    fun f(x: Int, n: Int): ModInt {
        if (x == 0 || x + n == W - 1) {
            return dp1[W - 1 - n]
        }

        return dp1[x] * dp1[W - n - x - 1]
    }

    for (i in 1..H) {
        for (j in 0 until W) {
            dp[i][j] += dp[i - 1][j] * f(j, 0)

            if (j > 0) {
                dp[i][j] += dp[i - 1][j - 1] * f(j - 1, 1)
            }

            if (j < W - 1) {
                dp[i][j] += dp[i - 1][j + 1] * f(j, 1)
            }
        }
    }

    println(dp[H][K - 1].value)
}

class ModInt (value: Long, val mod: Long) {
    init {
        if (mod <= 0) {
            throw IllegalArgumentException("ModInt cannot be created for non-positive modulus: $mod")
        }
    }

    val value = (value % mod + mod) % mod

    private fun assertSameMod(other: ModInt) {
        if (mod != other.mod) {
            throw IllegalArgumentException(
                "You cannot operate summation, subtraction and multiplication for ModInts with different moduli: $mod, ${other.mod}"
            )
        }
    }

    operator fun unaryMinus() = ModInt(mod - value, mod)

    operator fun plus(other: ModInt): ModInt {
        assertSameMod(other)

        return ModInt((value + other.value) % mod, mod)
    }

    operator fun plus(other: Long) = plus(ModInt(other, mod))

    operator fun minus(other: ModInt): ModInt {
        assertSameMod(other)

        return plus(-other)
    }

    operator fun minus(other: Long) = minus(ModInt(other, mod))

    operator fun times(other: ModInt): ModInt {
        assertSameMod(other)

        return ModInt((value * other.value) % mod, mod)
    }

    operator fun times(other: Long) = times(ModInt(other, mod))

    fun pow(k: Long): ModInt {
        var res = 1L
        var a = value
        var l = k

        while (l > 0) {
            if ((l and 1L) == 1L) {
                res = (res * a) % mod
            }

            a = a * a % mod
            l /= 2L
        }

        return ModInt(res, mod)
    }

    /**
     * Returns an inverse element of the value.
     * To ensure `m * m.inv() = 1`, the modulus should be prime, since it uses Fermat's little theorem.
     */
    fun inv() = pow(mod - 2)

    operator fun div(other: ModInt): ModInt {
        assertSameMod(other)
        if (other.value == 0L) {
            throw ArithmeticException()
        }

        return times(other.inv())
    }

    operator fun div(other: Long) = div(ModInt(other, mod))
}