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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
#define INF 2147483647
#define INF_LL 9223372036854775807
#define MOD 1000000007
using namespace std;
typedef long long int ll;
typedef unsigned long long int ull;

vector<ll> next(int pos, int W) {
	vector<ll> ans(3, 0);
	for (int i = 0; i < (1 << (W-1)); i++) {
		bool flag = false;
		for (int j = 1; j < W - 1; j++) {
			if (i & (1 << (j - 1)) && i & (1 << j)) {
				flag = true;
				break;
			}
		}
		if (flag)continue;
		if (i & (1 << (pos - 1)))ans[0]++;
		else if (i & (1 << pos))ans[2]++;
		else ans[1]++;
	}
	return ans;
}
int main() {
	int H, W, K;
	cin >> H >> W >> K;
	vector<vector<ll> > dp(H+1, vector<ll>(W, 0));
	dp[0][0] = 1;
	for (int i = 0; i < H; i++) {
		for (int j = 0; j < W; j++) {
			dp[i][j] %= MOD;
			auto a = next(j, W);
			if (j != 0)dp[i + 1][j - 1] += dp[i][j] * a[0];
			dp[i + 1][j] += dp[i][j] * a[1];
			if (j != W - 1)dp[i + 1][j + 1] = dp[i][j] * a[2];
		}
	}
	cout << dp[H][K - 1] % MOD << endl;
	return 0;
}