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;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Text;
using System.Threading.Tasks;
using static System.Console;
using static System.Math;
class Z { static void Main() => new K(); }
class K
{
	int F => int.Parse(Str);
	long FL => long.Parse(Str);
	int[] G => Strs.Select(int.Parse).ToArray();
	long[] GL => Strs.Select(long.Parse).ToArray();
	string Str => ReadLine();
	string[] Strs => Str.Split(new char[] { ' ', '\t' }, StringSplitOptions.RemoveEmptyEntries);
	const int MOD = 1000000007;
	public K()
	{
		SetOut(new StreamWriter(OpenStandardOutput()) { AutoFlush = false });
		Solve();
		Out.Flush();
	}
	void Solve()
	{
		var I = G;
		int H = I[0], W = I[1], K = I[2] - 1;
		if (W == 1) { WriteLine(1); return; }
		var A = new long[W, W];
		for (var i = 0; i < 1 << (W - 1); i++)
		{
			var flag = true;
			for (var j = 0; j < W - 1; j++) flag &= (i & (3 << j)) != (3 << j);
			if (!flag) continue;
			for (var j = 0; j < W - 1; j++)
				if ((i & (1 << j)) == 0) A[j, j]++;
				else
				{
					A[j, j + 1]++;
					A[j + 1, j]++;
					j++;
				}
			if ((i & (1 << (W - 2))) == 0) A[W - 1, W - 1]++;
		}
		var M = new Matrix(A);
		M = M.Pow((ulong)H, MOD);
		WriteLine(M[0, K]);
	}
}
class Matrix
{
	public int Row { get; protected set; }
	public int Column { get; protected set; }
	private long[,] a;
	public long this[int r, int c] { get { return a[r, c]; } set { a[r, c] = value; } }
	public Matrix(int r, int c) { Row = r; Column = c; a = new long[r, c]; }
	public Matrix(long[,] components) : this(components.GetLength(0), components.GetLength(1), (i, j) => components[i, j]) { }
	public Matrix(int r, int c, Func<int, int, long> components) : this(r, c) { for (var i = 0; i < r; i++) for (var j = 0; j < c; j++) this.a[i, j] = components(i, j); }
	public static Matrix operator +(Matrix p) => new Matrix(p.a);
	public static Matrix operator %(Matrix p, long mod) => new Matrix(p.Row, p.Column, (i, j) => p.a[i, j] % mod);
	public static Matrix operator *(Matrix p, Matrix q)
		=> new Matrix(p.Row, q.Column, (i, j) => { var sum = 0L; for (var k = 0; k < p.Column; k++) sum = (sum + p.a[i, k] * q.a[k, j]); return sum; });
	public static Matrix One(int size) => new Matrix(size, size, (i, j) => i == j ? 1 : 0);
	public Matrix Pow(ulong n, long mod)
	{
		var p = One(Row);
		var x = this % mod;
		while (n > 0)
		{
			if ((n & 1) == 1) p = (p * x) % mod;
			n >>= 1;
			x = (x * x) % mod;
		}
		return p % mod;
	}
}