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.Linq;
using static System.Console;
class Z { static void Main() => new K(); }
class K
{
	int[] G => ReadLine().Split().Select(int.Parse).ToArray();
	public const int MOD = 1000000007;
	public K()
	{
		var I = G;
		int H = I[0], W = I[1], K = I[2] - 1;
		if (W == 1) { WriteLine(1); return; }
		var fib = new int[W + 1];
		fib[1] = fib[2] = 1;
		for (var n = 3; n <= W; n++) fib[n] = (fib[n - 1] + fib[n - 2]) % MOD;
		var M = new Matrix(W);
		for (var i = 0; i < W; i++) M[i, i] = fib[i + 1] * fib[W - i] % MOD;
		for (var i = 0; i < W - 1; i++) M[i, i + 1] = M[i + 1, i] = fib[i + 1] * fib[W - i - 1] % MOD;
		WriteLine(M.Pow(H)[0, K]);
	}
}
class Matrix
{
	const long MOD = K.MOD;
	public readonly int Size;
	readonly long[,] a;
	public long this[int r, int c] { get { return a[r, c]; } set { a[r, c] = value; } }
	public Matrix(int s) { Size = s; a = new long[s, s]; }
	public Matrix(int s, Func<int, int, long> f) : this(s) { for (var i = 0; i < s; i++) for (var j = 0; j < s; j++) a[i, j] = f(i, j); }
	public static Matrix operator *(Matrix p, Matrix q)
		=> new Matrix(p.Size, (i, j) => { var sum = 0L; for (var k = 0; k < p.Size; k++) sum = (sum + p.a[i, k] * q.a[k, j] % MOD) % MOD; return sum; });
	public static Matrix One(int size) => new Matrix(size, (i, j) => i == j ? 1 : 0);
	public Matrix Pow(long n)
	{
		var p = One(Size);
		var x = this;
		while (n > 0)
		{
			if ((n & 1) == 1) p *= x;
			n >>= 1;
			x *= x;
		}
		return p;
	}
}