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# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.IO;
using System.Collections.Generic;
using System.Linq;

namespace AtCoder
{
    class Program
    {
        static void Main(string[] args)
        {
#if DEBUG
            Solve(Console.In, Console.Out);
            Console.ReadLine();
#else
            Solve(Console.In, Console.Out);
#endif
        }

        public static void Solve(TextReader ci, TextWriter co)
        {
            int h, w, k;
            ReadInts(ci, out h, out w, out k);
            co.WriteLine(Solve(h, w, 0, k-1));
            co.Flush();
        }

        private static long Solve(int h, int w, int start, int goal)
        {
            int d = Math.Abs(start - goal);
            if (d > h) {
                return 0;
            }

            int key = (h << 16) | (start << 4) | (goal);
            long numTotalAns = 0;
            if (memo1.TryGetValue(key , out numTotalAns)) {
                return numTotalAns;
            }

            else if (h == 1) {
                int min = Math.Min(start, goal);
                int max = Math.Max(start, goal);
                return BarPattern(min - 1) * BarPattern(w - max - 2);
            }
            else {
                int h0 = h / 2;
                int h1 = h - h0;
                numTotalAns = 0;
                for (int mid = 0; mid < w; ++mid) {
                    long numAns =
                        Solve(h0, w, start, mid) *
                        Solve(h1, w, mid, goal);
                    numTotalAns = (numTotalAns + numAns) % 1000000007;
                }
                memo1.Add(key, numTotalAns);
                return numTotalAns;
            }
        }

        private static Dictionary<int, long> memo1 = new Dictionary<int, long>();


        private static long BarPattern(int n)
        {
            int numPattern;
            if (n <= 0) {
                return 1;
            }
            else if (memo2.TryGetValue(n, out numPattern)) {
                return numPattern;
            }
            else {
                numPattern = 0;
                int max = 1 << n;
                for (int i = 0; i < max; ++i) {
                    if ((i & (i << 1)) == 0) ++numPattern;
                }
                memo2.Add(n, numPattern);
                return numPattern;
            }
        }

        private static Dictionary<int, int> memo2 = new Dictionary<int, int>();

        public static int ReadInt(TextReader ci) => int.Parse(ci.ReadLine());

        public static void ReadInts(TextReader ci, out int a, out int b)
        {
            var array = IntArrayFrom(ci.ReadLine());
            a = array[0];
            b = array[1];
        }

        public static void ReadInts(TextReader ci, out int a, out int b, out int c)
        {
            var array = IntArrayFrom(ci.ReadLine());
            a = array[0];
            b = array[1];
            c = array[2];
        }

        public static int[] IntArrayFrom(string s)
            => s.Split(' ').Select(p => int.Parse(p)).ToArray();
    }
}