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 PyPy3 (2.4.0) (judge verdict: Accepted)

from sys import exit, setrecursionlimit, stderr
from functools import reduce
from itertools import *
from collections import defaultdict, Counter
from bisect import bisect
import functools

def read():
  return int(input())

def reads():
  return [int(x) for x in input().split()]

H, W, K = reads()
P = 10**9 + 7

M = W + 100
fib = [0] * M
fib[0] = fib[1] = 1
for i in range(M-2):
  fib[i+2] = (fib[i] + fib[i+1]) % P

@functools.lru_cache(100000)
def solve(h, k):
  if h == 0:
    return 1 if k == 1 else 0
  if k == 1:
    return (solve(h-1, 1) * fib[W-1] + solve(h-1, 2) * fib[W-2]) % P
  elif k == W:
    return (solve(h-1, W) * fib[W-1] + solve(h-1, W-1) * fib[W-2]) % P
  else:
    return (solve(h-1, k) * fib[k-1] * fib[W-k]
      + solve(h-1, k-1) * fib[k-2] * fib[W-k]
      + solve(h-1, k+1) * fib[k-1] * fib[W-k-1]) % P

if W == 1:
  print(1); exit()
print(solve(H, K))