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 Ruby (2.3.3) (judge verdict: Accepted)

# https://atcoder.jp/contests/abc113/tasks/abc113_d
# あみだくじは, 日本に古くから伝わる伝統的なくじ引きである.
# あみだくじを作るには, まず W 本の平行な縦線を引き, 次にそれらを繋ぐ横線を引いていく.
# それぞれの縦棒の長さは H+1 [cm] であり、
# 横線の端点となれるのは上から 1,2,3,...,H [cm] の位置のみである.
# ここで,「正しいあみだくじ」とは, 以下のような条件を満たすあみだくじのことである.
# どの 2 つの横棒も端点を共有しない.
# それぞれの横棒の 2 つの端点は同じ高さになければならない.
# 横棒は隣り合う縦線を繋がなければならない.

# 縦棒 1 の上端から, 横線があれば必ずそれを通るというルールで下へたどったときに,
#  最終的にたどり着く縦棒の番号が K となるような「正しいあみだくじ」の本数を 1 000 000 007 で割った余りを求めなさい.
# H は 1 以上 100 以下の整数
# W は 1 以上 8 以下の整数
# K は 1 以上 W 以下の整数
# H W K
# 15 8 5
# 437760187
H, W, K = gets.split.map(&:to_i)

FIB = [1, 1, 2, 3, 5, 8, 13, 21]
MOD = (10**9) + 7

def getc(a,b)
  FIB[a] * FIB[W-1-b]
end

dp = Array.new(H+1) { Array.new(W, 0) }
dp[0][0] = 1
1.step(H) do |h|
  W.times do |w|
    dp[h][w]   += (dp[h-1][w] * getc(w,   w))%MOD
    dp[h][w-1] += (dp[h-1][w] * getc(w-1, w))%MOD if w != 0
    dp[h][w+1] += (dp[h-1][w] * getc(w, w+1))%MOD if w != W-1
  end
end
p dp[H][K-1]%MOD