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;
using System.Collections.Generic;
using System.Diagnostics;
using System.Globalization;
using System.IO;
using System.Text;
using System.Linq;

using E = System.Linq.Enumerable;

internal partial class Solver {
    bool Check(int bit) {
        while (bit > 0) {
            if (bit % 4 == 3) return false;
            bit >>= 1;
        }
        return true;
    }

    public void Run() {
        var h = ni();
        var w = ni(); // 1-8
        var k = ni();
        int mod = ModInt.SetModValue(1000000007);

        var mat = new ModMatrix(w, w);

        for (int b = 0; b < (1 << (w - 1)); b++) {
            if (!Check(b)) continue;
            var swapped = new bool[w];
            for (int j = 0; j < w - 1; j++) {
                if ((b & (1 << j)) != 0) { // swap j and j+1
                    mat[j, j + 1]++;
                    mat[j + 1, j]++;
                    swapped[j] = swapped[j + 1] = true;
                }
            }
            for (int j = 0; j < w; j++) {
                if (!swapped[j]) {
                    mat[j, j]++;
                }
            }
        }

        var dp = new ModInt[w];
        dp[0] = 1;
        var ans = ModMatrix.Pow(mat, h) * ModMatrix.TransposeFrom(dp);

        cout.WriteLine(ans[k - 1, 0]);
    }
}


public struct ModInt : IEquatable<ModInt> {
    private static int mod = 0;
    private long _value;

    public ModInt(long x) {
        _value = x;
        Normalize();
    }

    private static long RegularMod(long x, int mod) {
        if (x >= mod) {
            if (x < 2 * mod) {
                return x - mod;
            }

            return x % mod;
        }
        if (x >= 0) {
            return x;
        }

        x = mod - RegularMod(-x, mod);
        if (x == mod) {
            return 0;
        }

        return x;
    }

    public static int SetModValue(int m) {
        return mod = m;
    }

    private void Normalize() {
        _value = RegularMod(_value, mod);
    }

    public override string ToString() {
        return _value.ToString(CultureInfo.InvariantCulture);
    }

    public int ToInt() {
        return (int)_value;
    }

    public static bool operator ==(ModInt c1, ModInt c2) {
        return c1._value == c2._value;
    }

    public static bool operator !=(ModInt c1, ModInt c2) {
        return !(c1 == c2);
    }

    public static ModInt operator +(ModInt x, ModInt y) {
        return new ModInt(x._value + y._value);
    }
    public static ModInt operator -(ModInt x, ModInt y) {
        return new ModInt(x._value - y._value);
    }
    public static ModInt operator -(ModInt x) {
        return new ModInt(-x._value);
    }
    public static ModInt operator *(ModInt x, ModInt y) {
        return new ModInt(x._value * y._value);
    }
    public static ModInt operator /(ModInt x, ModInt y) {
        return new ModInt(x._value * Inverse(y._value, mod));
    }
    public static ModInt operator ++(ModInt x) {
        return x + 1;
    }
    public static ModInt operator --(ModInt x) {
        return x - 1;
    }
    public static ModInt Pow(ModInt x, long n) {
        ModInt r = 1;
        while (n > 0) {
            if ((n & 1) != 0) {
                r *= x;
            }

            x *= x;
            n >>= 1;
        }
        return r;
    }
    private static long ExtendedGcd(long a, long b, ref long x, ref long y) {
        if (b == 0) {
            x = 1; y = 0;
            return a;
        } else {
            var d = ExtendedGcd(b, a % b, ref y, ref x);
            y -= a / b * x;
            return d;
        }
    }

    private static long Inverse(long a, long mod) {
        long x = 0, y = 0;
        if (ExtendedGcd(a, mod, ref x, ref y) == 1) {
            return (x + mod) % mod;
        } else {
            throw new Exception("Invalid inverse " + a + " " + mod);
        }
    }

    public static implicit operator ModInt(long x) {
        return new ModInt(x);
    }

    public override bool Equals(object obj) {
        if (obj == null) {
            return false;
        }
        return _value.Equals(((ModInt)obj)._value);
    }

    public override int GetHashCode() {
        return _value.GetHashCode();
    }

    public bool Equals(ModInt other) {
        return _value.Equals(other._value);
    }
}

public class ModMatrix {
    public int Row { get; private set; }
    public int Col { get; private set; }
    private ModInt[] Data;
    public ModMatrix(int row, int col) {
        Row = row;
        Col = col;
        Data = new ModInt[Row * Col];
    }

    public ModMatrix(ModInt[,] array)
        : this(array.GetLength(0), array.GetLength(1)) {
        var index = 0;
        for (var i = 0; i < Row; i++) {
            for (var j = 0; j < Col; j++) {
                Data[index++] = array[i, j];
            }
        }
    }

    public static ModMatrix TransposeFrom(IReadOnlyList<ModInt> list) {
        var count = list.Count;
        var result = new ModInt[count, 1];
        for (int i = 0; i < count; i++) {
            result[i, 0] = list[i];
        }
        return new ModMatrix(result);
    }

    public ModInt this[int row, int col] {
        set {
            Data[row * Col + col] = value;
        }
        get {
            return Data[row * Col + col];
        }
    }

    public static ModMatrix UnitMatrix(int n) {
        var matrix = new ModMatrix(n, n);
        for (var i = 0; i < n; i++) {
            matrix[i, i] = 1;
        }
        return matrix;
    }

    public static ModMatrix operator +(ModMatrix A, ModMatrix B) {
        var C = new ModMatrix(A.Row, A.Col);
        for (var i = 0; i < C.Data.Length; i++) {
            C.Data[i] = A.Data[i] + B.Data[i];
        }
        return C;
    }

    public static ModMatrix operator -(ModMatrix A, ModMatrix B) {
        var C = new ModMatrix(A.Row, A.Col);
        for (var i = 0; i < C.Data.Length; i++) {
            C.Data[i] = A.Data[i] - B.Data[i];
        }
        return C;
    }

    public static ModMatrix operator *(ModMatrix A, ModMatrix B) {
        var C = new ModMatrix(A.Row, B.Col);
        for (var i = 0; i < A.Row; i++) {
            for (var j = 0; j < B.Col; j++) {
                var val = C[i, j];
                for (var k = 0; k < A.Col; k++) {
                    val += A[i, k] * B[k, j];
                }
                C[i, j] = val;
            }
        }
        return C;
    }

    public static ModMatrix Pow(ModMatrix A, long n) {
        if (n == 0) {
            return UnitMatrix(A.Row);
        }

        if (n == 1) {
            return A;
        }

        ModMatrix result = A.Clone(), t = A.Clone(); n--;
        while (n > 0) {
            if ((n & 1) != 0) {
                result = result * t;
            }
            t = t * t;
            n >>= 1;
        }

        return result;
    }

    public ModMatrix Clone() {
        return new ModMatrix(Row, Col) { Data = Data.Clone() as ModInt[] };
    }

    private static void Swap<T>(ref T a, ref T b) { var t = a; a = b; b = t; }

    public static ModInt Determinant(ModMatrix matrix) {
        if (matrix.Row != matrix.Col) {
            throw new ArgumentException();
        }

        var n = matrix.Col;

        var m = new ModInt[n][];
        for (var i = 0; i < n; i++) {
            var r = new ModInt[n];
            for (var j = 0; j < n; j++) {
                r[j] = matrix[i, j];
            }
            m[i] = r;
        }

        ModInt d = 1;

        for (var i = 0; i < n; i++) {
            var targetRow = -1;
            ModInt pivot = 0;
            for (var r = i; r < n; r++) {
                var row = m[r];
                if (row[i] != 0) {
                    targetRow = r;
                    pivot = row[i];
                    break;
                }
            }
            if (targetRow == -1) {
                return 0;
            }

            if (i != targetRow) {
                Swap(ref m[i], ref m[targetRow]);
                d = -d;
            }
            d *= pivot;
            var mi = m[i];
            var invPivot = 1 / pivot;
            for (var r = i + 1; r < n; r++) {
                var row = m[r];
                var weight = row[i] * invPivot;
                if (weight == 0) continue;
                for (var k = i + 1; k < n; k++) {
                    row[k] -= weight * mi[k];
                }
            }
        }

        return d;
    }
}
// PREWRITEN CODE BEGINS FROM HERE
internal partial class Solver : Scanner {
    public static void Main() {
#if LOCAL
        byte[] inputBuffer = new byte[1000000];
        var inputStream = Console.OpenStandardInput(inputBuffer.Length);
        using (var reader = new StreamReader(inputStream, Console.InputEncoding, false, inputBuffer.Length)) {
            Console.SetIn(reader);
            new Solver(Console.In, Console.Out).Run();
        }
#else
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Solver(Console.In, Console.Out).Run();
        Console.Out.Flush();
#endif
    }

#pragma warning disable IDE0052
    private readonly TextReader cin;
    private readonly TextWriter cout;
#pragma warning restore IDE0052

    public Solver(TextReader reader, TextWriter writer)
        : base(reader) {
        cin = reader;
        cout = writer;
    }
    public Solver(string input, TextWriter writer)
        : this(new StringReader(input), writer) {
    }

#pragma warning disable IDE1006
#pragma warning disable IDE0051
    private int ni() { return NextInt(); }
    private int[] ni(int n) { return NextIntArray(n); }
    private long nl() { return NextLong(); }
    private long[] nl(int n) { return NextLongArray(n); }
    private double nd() { return NextDouble(); }
    private double[] nd(int n) { return NextDoubleArray(n); }
    private string ns() { return Next(); }
    private string[] ns(int n) { return NextArray(n); }
#pragma warning restore IDE1006
#pragma warning restore IDE0051
}

internal static class LinqPadExtension {
    [Conditional("DEBUG")]
    public static void Dump<T>(this T obj) {
#if DEBUG
        LINQPad.Extensions.Dump(obj);
#endif
    }
}

public class Scanner {
    private readonly TextReader Reader;
    private readonly CultureInfo ci = CultureInfo.InvariantCulture;

    private readonly char[] buffer = new char[2 * 1024];
    private int cursor = 0, length = 0;
    private string Token;
    private readonly StringBuilder sb = new StringBuilder(1024);

    public Scanner()
        : this(Console.In) {
    }

    public Scanner(TextReader reader) {
        Reader = reader;
    }

    public int NextInt() { return checked((int)NextLong()); }
    public long NextLong() {
        var s = Next();
        long r = 0;
        int i = 0;
        bool negative = false;
        if (s[i] == '-') {
            negative = true;
            i++;
        }
        for (; i < s.Length; i++) {
            r = r * 10 + (s[i] - '0');
#if DEBUG
            if (!char.IsDigit(s[i])) throw new FormatException();
#endif
        }
        return negative ? -r : r;
    }
    public double NextDouble() { return double.Parse(Next(), ci); }
    public string[] NextArray(int size) {
        string[] array = new string[size];
        for (int i = 0; i < size; i++) {
            array[i] = Next();
        }

        return array;
    }
    public int[] NextIntArray(int size) {
        int[] array = new int[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextInt();
        }

        return array;
    }

    public long[] NextLongArray(int size) {
        long[] array = new long[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextLong();
        }

        return array;
    }

    public double[] NextDoubleArray(int size) {
        double[] array = new double[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextDouble();
        }

        return array;
    }

    public string Next() {
        if (Token == null) {
            if (!StockToken()) {
                throw new Exception();
            }
        }
        var token = Token;
        Token = null;
        return token;
    }

    public bool HasNext() {
        if (Token != null) {
            return true;
        }

        return StockToken();
    }

    private bool StockToken() {
        while (true) {
            sb.Clear();
            while (true) {
                if (cursor >= length) {
                    cursor = 0;
                    if ((length = Reader.Read(buffer, 0, buffer.Length)) <= 0) {
                        break;
                    }
                }
                var c = buffer[cursor++];
                if (33 <= c && c <= 126) {
                    sb.Append(c);
                } else {
                    if (sb.Length > 0) break;
                }
            }

            if (sb.Length > 0) {
                Token = sb.ToString();
                return true;
            }

            return false;
        }
    }
}