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.Collections.Generic;
using System.Linq;
using BitArray = System.Collections.BitArray;
using BigInteger = System.Numerics.BigInteger;

namespace AtCoderProject
{
    public class Program
    {
        public object Calc()
        {
            var line = consoleReader.SplitInt32();
            var H = line[0];
            var W = line[1];
            var K = line[2];
            var dp = new long[H + 1, W + 1];

            dp[0, 1] = 1;

            for (int h = 1; h <= H; h++)
            {
                for (int w = 1; w <= W; w++)
                {
                    long left, right, center;

                    if (w - 1 <= 0) left = 0;
                    else if (w - 1 == 1 || w == W) left = honsu[W - 2] * dp[h - 1, w - 1];
                    else left = honsu[w - 2] * honsu[W - w] * dp[h - 1, w - 1];

                    if (w + 1 > W) right = 0;
                    else if (w + 1 == W || w == 1) right = honsu[W - 2] * dp[h - 1, w + 1];
                    else right = honsu[w - 1] * honsu[W - w - 1] * dp[h - 1, w + 1];

                    if (w == 1 || w == W) center = honsu[W - 1] * dp[h - 1, w];
                    else center = honsu[w - 1] * honsu[W - w] * dp[h - 1, w];

                    dp[h, w] = (left + center + right) % mod;
                }
            }

            return dp[H, K] ;
        }

        static long[] honsu = new long[9];
        static Program()
        {
            honsu[0] = honsu[1] = 1;
            long prev = honsu[2] = 2;
            long rightEmpty = 1;
            for (int l = 3; l <= 8; l++)
            {
                honsu[l] = prev + rightEmpty;
                rightEmpty = prev;
                prev = honsu[l];
            }
        }

        public const int mod = 1000000007;
        private readonly ConsoleReader consoleReader;
        public Program() : this(new ConsoleReader()) { }
        public Program(ConsoleReader consoleReader) { this.consoleReader = consoleReader; }
        static void Main() => Console.WriteLine(new Program().Calc());
        public static string MultiLine<T>(IEnumerable<T> source) => string.Join("\n", source);
    }

    public class ConsoleReader { public virtual string String() => Console.ReadLine(); public int Int32() => int.Parse(String()); public long Int64() => long.Parse(String()); public ulong UInt64() => ulong.Parse(String()); public double Double() => double.Parse(String()); public string[] Split() => String().Split().ToArray(); public int[] SplitInt32() => String().Split().Select(int.Parse).ToArray(); public long[] SplitInt64() => String().Split().Select(long.Parse).ToArray(); public ulong[] SplitUInt64() => String().Split().Select(ulong.Parse).ToArray(); public double[] SplitDouble() => String().Split().Select(double.Parse).ToArray(); public IEnumerable<string> Repeat(int count) { for (int i = 0; i < count; i++) yield return String(); } public IEnumerable<int> RepeatInt32(int count) { for (int i = 0; i < count; i++) yield return Int32(); } public IEnumerable<long> RepeatInt64(int count) { for (int i = 0; i < count; i++) yield return Int64(); } public IEnumerable<string[]> RepeatSplit(int count) { for (int i = 0; i < count; i++) yield return Split(); } public IEnumerable<int[]> RepeatSplitInt32(int count) { for (int i = 0; i < count; i++) yield return SplitInt32(); } }
}