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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace AtCoder
{
    class Program
    {
        private static string Read() { return Console.ReadLine(); }
        private static int ReadInt() { return int.Parse(Read()); }
        private static long ReadLong() { return long.Parse(Read()); }
        private static int[] ReadInts() { return Array.ConvertAll(Read().Split(), int.Parse); }
        private static long[] ReadLongs() { return Array.ConvertAll(Read().Split(), long.Parse); }

        static long Solve(int H, int W, int K)
        {
            const int M = 1000000007;
            var counts = new int[W][];
            for (int i = 0; i < W; ++i) { counts[i] = new int[W]; }
            for (int i = 0; i < (1 << (W - 1)); ++i)
            {
                bool valid = true;
                for (int j = 0; j < W - 1; ++j)
                {
                    int mask = 3 << j;
                    if ((i & mask) == mask) { valid = false; }
                }
                if (valid)
                {
                    for (int j = 0; j < W; ++j)
                    {
                        int L = (j - 1 >= 0) ? 1 << (j - 1) : 0;
                        int R = (j < W - 1) ? 1 << j : 0;
                        if ((i & L) != 0)
                        {
                            ++counts[j][j - 1];
                        }
                        else if ((i & R) != 0)
                        {
                            ++counts[j][j + 1];
                        }
                        else
                        {
                            ++counts[j][j];
                        }
                    }
                }
            }
            var sum1 = new long[W];
            sum1[0] = 1;
            for (int i = 0; i < H; ++i)
            {
                var sum2 = new long[W];
                for (int j = 0; j < W; ++j)
                {
                    for (int k = 0; k < W; ++k)
                    {
                        sum2[k] += (sum1[j] * counts[j][k]) % M;
                        sum2[k] %= M;
                    }
                }
                sum1 = sum2;
            }
            return sum1[K - 1];
        }
        static void Main(string[] args)
        {
            var V = ReadInts();
            Console.WriteLine(Solve(V[0], V[1], V[2]));
        }
    }
}