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.util.Scanner;

public class Main {
    private static final long MOD = 1_000_000_007;
    private int sum = 0;

    public void main(Scanner sc) {
        int h = sc.nextInt();
        int w = sc.nextInt();
        int k = sc.nextInt();

        int cnt[] = new int[w + 1];
        init(w, cnt);

        long dp[][] = new long[h + 1][w];
        dp[0][0] = 1;
        for (int i = 1; i <= h; i++) {
            for (int j = 0; j < w; j++) {
                // 左
                if (j > 0) {
                    dp[i][j - 1] += cnt[j] * dp[i - 1][j];
                    dp[i][j - 1] %= MOD;
                }

                // 中
                dp[i][j] += (sum - cnt[j] - cnt[j + 1]) * dp[i - 1][j];
                dp[i][j] %= MOD;

                // 右
                if (j + 1 < w) {
                    dp[i][j + 1] += cnt[j + 1] * dp[i - 1][j];
                    dp[i][j + 1] %= MOD;
                }
            }
        }

        System.out.println(dp[h][k - 1]);
    }

    private void init(int w, int cnt[]) {
        for (int n = 0; n < (1 << (w - 1)); n++) {
            int tmp = n;
            int bi[] = new int[w];
            boolean ok = true;
            for (int i = 0; i < w - 1; i++) {
                bi[i + 1] = tmp % 2;
                if (bi[i] * bi[i + 1] == 1) {
                    ok = false;
                    break;
                }
                tmp /= 2;
            }

            if (ok) {
                sum++;
                for (int i = 1; i < w; i++) {
                    cnt[i] += bi[i];
                }
            }
        }
    }

    public static void main(String[] args) {
        try (Scanner sc = new Scanner(System.in)) {
            new Main().main(sc);
        } catch (Exception e) {
            throw e;
        }
    }
}