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 Yorukatsu020.Questions;
using Yorukatsu020.Extensions;
using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Text;

namespace Yorukatsu020.Questions
{
    public class QuestionE : AtCoderQuestionBase
    {
        public override IEnumerable<object> Solve(TextReader inputStream)
        {
            var hwk = inputStream.ReadIntArray();
            var h = hwk[0];
            var w = hwk[1];
            var k = hwk[2];
            var candidate = w - 1;
            var count = new Modular[h + 1, w];
            ClearArray(count);
            count[0, 0] = new Modular(1);

            for (int row = 1; row <= h; row++)
            {
                for (int column = 0; column < w; column++)
                {
                    int leftPatterns;
                    int rightPatterns;

                    // 左から来るやつ
                    if (column > 0)
                    {
                        leftPatterns = ColumnwiseDP(column - 1, true);
                        rightPatterns = ColumnwiseDP(w - column - 2, false);
                        count[row, column] += count[row - 1, column - 1] * new Modular(leftPatterns * rightPatterns);
                    }

                    // 上からそのまま来るやつ
                    leftPatterns = ColumnwiseDP(column - 1, false);
                    rightPatterns = ColumnwiseDP(w - column - 2, false);
                    count[row, column] += count[row - 1, column] * new Modular(leftPatterns * rightPatterns);

                    // 右から来るやつ
                    if (column < w - 1)
                    {
                        leftPatterns = ColumnwiseDP(column - 1, false);
                        rightPatterns = ColumnwiseDP(w - column - 2, true);
                        count[row, column] += count[row - 1, column + 1] * new Modular(leftPatterns * rightPatterns);
                    }
                }
            }

            yield return count[h, k - 1].Value;
        }

        private int ColumnwiseDP(int length, bool first)
        {
            if (length <= 0)
            {
                return 1;
            }

            var dp = new int[length + 1, 2];
            if (first)
            {
                dp[0, 1] = 1;
            }
            else
            {
                dp[0, 0] = 1;
            }

            for (int column = 1; column <= length; column++)
            {
                // 繋がない
                dp[column, 0] += dp[column - 1, 0];
                dp[column, 0] += dp[column - 1, 1];

                // 繋ぐ
                dp[column, 1] += dp[column - 1, 0];     // 1個前で繋いでない場合のみ
            }

            return dp[length, 0] + dp[length, 1];
        }

        private void ClearArray(Modular[,] array)
        {
            for (int i = 0; i < array.GetLength(0); i++)
            {
                for (int j = 0; j < array.GetLength(1); j++)
                {
                    array[i, j] = new Modular(0);
                }
            }
        }

        static readonly int[] pow2Array = new int[] { 1, 2, 4, 8, 16, 32, 64, 128, 256 };
        private int Pow2(int n) => n > 0 ? pow2Array[n] : 1;

        [System.Runtime.InteropServices.StructLayout(System.Runtime.InteropServices.LayoutKind.Auto)]
        public struct Modular : IEquatable<Modular>, IComparable<Modular>
        {
            private const int _defaultMod = 1000000007;
            private readonly long _value;    // 0 <= value < Mod の値をとる
            public int Mod { get; }
            public int Value => (int)_value;

            public Modular(long value, int mod = _defaultMod)
            {
                if (mod < 2)
                {
                    throw new ArgumentOutOfRangeException(nameof(mod), $"{nameof(mod)}は2以上の素数でなければなりません。");
                }
                Mod = mod;

                if (value >= 0 && value < mod)
                {
                    _value = value;
                }
                else
                {
                    _value = value % mod;
                    if (Value < 0)
                    {
                        _value += mod;
                    }
                }
            }

            private static void CheckModEquals(Modular a, Modular b)
            {
                if (a.Mod != b.Mod)
                {
                    throw new ArgumentException($"{nameof(a)}, {nameof(b)}", $"両者の法（{nameof(Mod)}）は等しくなければなりません。");
                }
            }

            public static Modular operator +(Modular a, Modular b)
            {
                CheckModEquals(a, b);

                var result = a._value + b._value;
                if (result > a.Mod)
                {
                    result -= a.Mod;    // 剰余演算を避ける
                }
                return new Modular(result, a.Mod);
            }

            public static Modular operator -(Modular a, Modular b)
            {
                CheckModEquals(a, b);

                var result = a._value - b._value;
                if (result < 0)
                {
                    result += a.Mod;    // 剰余演算を避ける
                }
                return new Modular(result, a.Mod);
            }

            public static Modular operator *(Modular a, Modular b)
            {
                CheckModEquals(a, b);
                return new Modular(a._value * b._value, a.Mod);
            }

            public static Modular operator /(Modular a, Modular b)  // こいつは値渡しの方が速いっぽい
            {
                CheckModEquals(a, b);
                return a * Pow(b, a.Mod - 2);
            }

            // 需要は不明だけど一応
            public static bool operator ==(Modular left, Modular right) => left.Equals(right);
            public static bool operator !=(Modular left, Modular right) => !(left == right);
            public static bool operator <(Modular left, Modular right) => left.CompareTo(right) < 0;
            public static bool operator <=(Modular left, Modular right) => left.CompareTo(right) <= 0;
            public static bool operator >(Modular left, Modular right) => left.CompareTo(right) > 0;
            public static bool operator >=(Modular left, Modular right) => left.CompareTo(right) >= 0;

            public static explicit operator int(Modular a) => a.Value;
            public static explicit operator long(Modular a) => a._value;

            public static Modular Pow(Modular a, int n)
            {
                switch (n)
                {
                    case 0:
                        return new Modular(1, a.Mod);
                    case 1:
                        return a;
                    default:
                        var p = Pow(a, n / 2);
                        return p * p * Pow(a, n % 2);
                }
            }

            private static Dictionary<int, List<int>> FactorialCache { get; } = new Dictionary<int, List<int>>();
            private static Dictionary<int, List<int>> PermutationCache { get; } = new Dictionary<int, List<int>>();

            public static Modular Factorial(int n, int mod = _defaultMod)
            {
                if (n < 0)
                {
                    throw new ArgumentOutOfRangeException(nameof(n), $"{nameof(n)}は0以上の整数でなければなりません。");
                }
                if (mod < 2)
                {
                    throw new ArgumentOutOfRangeException(nameof(mod), $"{nameof(mod)}は2以上の素数でなければなりません。");
                }

                if (!FactorialCache.ContainsKey(mod))
                {
                    FactorialCache.Add(mod, new List<int>() { 1 });
                }

                var cache = FactorialCache[mod];
                for (int i = cache.Count; i <= n; i++)  // Countが1（0!までキャッシュ済み）のとき1!～n!まで計算
                {
                    cache.Add((int)((long)cache[i - 1] * i % mod));
                }
                return new Modular(cache[n], mod);
            }

            public static Modular Combination(int n, int r, int mod = _defaultMod)
            {
                CheckNR(n, r);
                return n == r ? new Modular(1, mod) : Factorial(n, mod) / (Factorial(r, mod) * Factorial(n - r, mod));   // nCr = n! / (n-r)!r! = nPr / n!
            }

            private static void CheckNR(int n, int r)
            {
                if (n <= 0)
                {
                    throw new ArgumentOutOfRangeException(nameof(n), $"{nameof(n)}は正の整数でなければなりません。");
                }
                if (r < 0)
                {
                    throw new ArgumentOutOfRangeException(nameof(r), $"{nameof(r)}は0以上の整数でなければなりません。");
                }
                if (n < r)
                {
                    throw new ArgumentOutOfRangeException($"{nameof(n)},{nameof(r)}", $"{nameof(r)}は{nameof(n)}以下でなければなりません。");
                }
            }

            public override string ToString() => $"{_value} (mod {Mod})";

            public bool Equals(Modular other) => _value == other._value && Mod == other.Mod;

            public override bool Equals(object obj) => obj is Modular && Equals((Modular)obj);

            public int CompareTo(Modular other) => Mod == other.Mod ? _value.CompareTo(other._value) : 0;

            public override int GetHashCode() => Value ^ Mod;
        }

    }
}


namespace Yorukatsu020
{
    class Program
    {
        static void Main(string[] args)
        {
            IAtCoderQuestion question = new QuestionE();    // 問題に合わせて書き換え
            var answers = question.Solve(Console.In);
            foreach (var answer in answers)
            {
                Console.WriteLine(answer);
            }
        }
    }
}

#region Base Classes

namespace Yorukatsu020.Questions
{

    public interface IAtCoderQuestion
    {
        IEnumerable<object> Solve(string input);
        IEnumerable<object> Solve(TextReader inputStream);
    }

    public abstract class AtCoderQuestionBase : IAtCoderQuestion
    {
        public IEnumerable<object> Solve(string input)
        {
            var stream = new MemoryStream(Encoding.Unicode.GetBytes(input));
            var reader = new StreamReader(stream, Encoding.Unicode);

            return Solve(reader);
        }

        public abstract IEnumerable<object> Solve(TextReader inputStream);
    }

}

#endregion

#region Extensions

namespace Yorukatsu020.Extensions
{
    internal static class TextReaderExtensions
    {
        internal static int ReadInt(this TextReader reader) => int.Parse(ReadString(reader));
        internal static long ReadLong(this TextReader reader) => long.Parse(ReadString(reader));
        internal static double ReadDouble(this TextReader reader) => double.Parse(ReadString(reader));
        internal static string ReadString(this TextReader reader) => reader.ReadLine();

        internal static int[] ReadIntArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(int.Parse).ToArray();
        internal static long[] ReadLongArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(long.Parse).ToArray();
        internal static double[] ReadDoubleArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(double.Parse).ToArray();
        internal static string[] ReadStringArray(this TextReader reader, char separator = ' ') => reader.ReadLine().Split(separator);
    }
}

#endregion