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 PHP (7.4.4) (judge verdict: Accepted)

<?php
const MOD = 10 ** 9 + 7;
list($H, $W, $K) = ints();
// 動的計画法
// dp[i][j]: i行目の移動を終えた後、j列目にいるあみだくじの本数
$dp = array_fill(0, $H + 1, array_fill(1, $W, 0));
$dp[0][1] = 1;
for ($i = 1; $i <= $H; ++$i) {
    // i行目の線の引き方を全探索
    for ($j = 0; $j < 2 ** ($W - 1); ++$j) {
        $next = array_fill(1, $W, 0);
        for ($k = 0; $k < $W - 1; ++$k) {
            if ($j >> $k & 1) {
                // 既に線がある場合は「正しいあみだくじ」ではないのでスキップ
                if ($next[$k + 1] !== 0) continue 2;
                $next[$k + 1] = 1; // 今の列から右に伸びる
                $next[$k + 2] = -1; // 右隣の列から左に伸びる
            }
        }
        for ($x = 1; $x <= $W; ++$x) {
            $dp[$i][$x + $next[$x]] += $dp[$i - 1][$x];
            $dp[$i][$x + $next[$x]] %= MOD;
        }
    }
}
echo $dp[$H][$K];
function ints()
{
    return array_map('intval', explode(' ', trim(fgets(STDIN))));
}