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)

require 'pp'
# IS_DEBUG = true
IS_DEBUG = false
def dputs str
	if(IS_DEBUG)
		puts str
	end
end
def dpp str
	if(IS_DEBUG)
		pp str
	end
end

require 'pp'

H,W,K=gets.chomp.split(' ').map{|n| n.to_i}
DP = Array.new(H){Array.new(W){0}} # 0 indexed
FIB = Array.new(W) # FIB[i] : upto i ladders
if(W==1)
	puts 1
	exit(0)
end
# precomputation
FIB[0] = 1
FIB[1] = 2
for i in 2..(W-1)
	FIB[i] = FIB[i-1] + FIB[i-2]
end
if(H==1)
	if(K==1)
		puts FIB[[W-2,0].max]
	elsif(K==2)
		puts FIB[[W-3,0].max]
	end
	exit(0)
end
# DP
DP[0][0] = FIB[W-2]
DP[0][1] = FIB[W-3]
for i in 1..(H-1)
	for j in 0..(W-1)
		DP[i][j] += DP[i-1][j-1] * FIB[[j-2,0].max] * FIB[[W-1-j-1,0].max] if j > 0
		DP[i][j] += DP[i-1][j]   * FIB[[j-1,0].max] * FIB[[W-1-j-1,0].max]
		DP[i][j] += DP[i-1][j+1] * FIB[[j-1,0].max] * FIB[[W-1-j-2,0].max] if j < W-1
		DP[i][j] %= 1000000007
	end
end
puts DP[H-1][K-1]

# pp DP