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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.List;
import java.util.Scanner;

public class Main {

    private static final long MOD = 1_000_000_007;

    private static int nextUnitAmida(int x) {
        int mask1 = 1;
        int mask2 = 3;
        ++x;
        for (;;) {
            int ck = x & mask2;
            if (ck == mask1) {
                break;
            } else if (ck == mask2) {
                x += mask1;
            } else {
                ;
            }
            mask1 <<= 1;
            mask2 <<= 1;
        }
        return x;
    }

    private static int[] createUnitAmida(int w) {
        List<Integer> uAmidas = new ArrayList<>();
        int x = 0;
        int limit = (1 << (w - 1));
        do {
            uAmidas.add(x);
            x = nextUnitAmida(x);
        } while (x < limit);

        int[] result = new int[uAmidas.size()];
        for (int i = 0; i < result.length; ++i) {
            result[i] = uAmidas.get(i);
        }
        return result;
    }

    private final int height;

    private final int width;

    private final int[] unitAmidas;

    private Main(int h, int w) {
        height = h;
        width = w;
        unitAmidas = createUnitAmida(w);
    }

    long countIt(int goal) {
        return countSub(height, 1, goal);
    }

    private long countSub(int h, int start, int goal) {
        if (Math.abs(start - goal) > h) {
            return 0;
        }
        if (h > 1) {
            final int h1 = h / 2;
            final int h2 = h - h1;
            long result = 0;
            for (int k = 0; k < width; ++k) {
                long r1 = countSub(h1, start, k + 1);
                long r2 = countSub(h2, k + 1, goal);
                result += r1 * r2;
            }
            return result % MOD;
        }

        assert h == 0;

        final int rMask = (1 << (start - 1));
        final int lMask = (rMask >> 1);

        int rCase = 0;
        int lCase = 0;
        int mCase = 0;

        for (int unitAmida : unitAmidas) {
            if ((unitAmida & rMask) != 0) {
                ++rCase;
            } else if ((unitAmida & lMask) != 0) {
                ++lCase;
            } else {
                ++mCase;
            }
        }

        long result = 0;
        if (start == goal) {
            result = mCase;
        } else if (start == goal - 1) {
            result = rCase;
        } else {
            result = lCase;
        }
        return result;
    }

    public static void main(String[] args) {
        try (
                Scanner in = new Scanner(System.in);
                PrintWriter out = new PrintWriter(System.out))
        {
            final int H = in.nextInt();
            final int W = in.nextInt();
            final int K = in.nextInt();

            long result = new Main(H, W).countIt(K);
            out.println(result);
        }
    }
}