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 <iostream>
#include <cstdio>
#include <cstring>
using namespace std;

typedef long long ll;
ll M = 1e9 + 7;
ll n, m, en, d[105][10], s[15];
ll g(ll p) {
	if (p < 1) return 1;
	else return s[p];
}
ll f(ll p, ll q) {
	ll r = 0;
	if (d[p][q] != -1) return d[p][q];
	if (p == n) return (q == en);
	r = (f(p + 1, q) * g(q - 2) * g(m - q - 1)) % M;
	if (q > 1) r = (r + f(p + 1, q - 1) * g(q - 3) * g(m - q - 1)) % M;
	if (q < m) r = (r + f(p + 1, q + 1) * g(q - 2) * g(m - q - 2)) % M;
	return d[p][q] = r;
}
int main()
{
	ll i;
	cin >> n >> m >> en;
	s[0] = 1;
	s[1] = 2;
	for (i = 2; i <= 10; i++) s[i] = s[i - 1] + s[i - 2];
	memset(d, -1, sizeof(d));
	cout << f(0, 1);
    return 0;
}