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: Wrong Answer)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using static System.Math;
using System.Text;

namespace Program
{
    public static class ABC113D
    {
        static public void Solve()
        {
            var H = NextInt;
            var W = NextInt;
            var K = NextInt;

            if (W == 1)
            {
                Console.WriteLine(1);
                return;
            }

            var dp = new Mod[H + 1, W];
            dp[0, 0] = 1;
            for (int i = 1; i < W; i++)
            {
                dp[0, i] = 0;
            }
            for (int i = 1; i <= H; i++)
            {
                for (int j = 0; j < W; j++)
                {
                    dp[i, j] = 0;
                    for (int k = 0; k < Pow(2, W - 1); k++)
                    {
                        var ok = true;
                        var mask = new[] { 1, 2, 4, 8, 16, 32, 63 };
                        for (int l = 0; l < W - 1; l++)
                        {
                            if (l == 0)
                            {
                                if ((k & mask[l]) != 0 && (k & mask[l + 1]) != 0)
                                {
                                    ok = false;
                                    break;
                                }
                            }
                            else if (l == W - 2)
                            {
                                if ((k & mask[l]) != 0 && (k & mask[l - 1]) != 0)
                                {
                                    ok = false;
                                    break;
                                }
                            }
                            else
                            {
                                if ((k & mask[l]) != 0 && ((k & mask[l - 1]) != 0 || (k & mask[l + 1]) != 0))
                                {
                                    ok = false;
                                    break;
                                }
                            }
                        }
                        if (ok)
                        {
                            if (j == 0)
                            {
                                if ((k & mask[0]) == 0)
                                {
                                    dp[i, j] += dp[i - 1, j];
                                }
                                else
                                {
                                    dp[i, j] += dp[i - 1, j + 1];
                                }
                            }
                            else if (j == W - 1)
                            {
                                if ((k & mask[j - 1]) == 0)
                                {
                                    dp[i, j] += dp[i - 1, j];
                                }
                                else
                                {
                                    dp[i, j] += dp[i - 1, j - 1];
                                }
                            }
                            else
                            {
                                if ((k & mask[j]) == 0 && (k & mask[j - 1]) == 0)
                                {
                                    dp[i, j] += dp[i - 1, j];
                                }
                                else if ((k & mask[j]) != 0)
                                {
                                    dp[i, j] += dp[i - 1, j + 1];
                                }
                                else
                                {
                                    dp[i, j] += dp[i - 1, j - 1];
                                }
                            }
                        }
                    }
                }
            }

            Console.WriteLine(dp[H, K - 1]);
        }

        static public void Main(string[] args)
        {
            if (args.Length == 0)
            {
                var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
                Console.SetOut(sw);
            }
            Solve();
            Console.Out.Flush();
        }
        static Random rand = new Random();
        static class Console_
        {
            private static Queue<string> param = new Queue<string>();
            public static string NextString()
            {
                if (param.Count == 0) foreach (var item in Console.ReadLine().Split(' ')) param.Enqueue(item);
                return param.Dequeue();
            }
        }
        static int NextInt => int.Parse(Console_.NextString());
        static long NextLong => long.Parse(Console_.NextString());
        static double NextDouble => double.Parse(Console_.NextString());
        static string NextString => Console_.NextString();
        static List<int> NextIntList(int N) => Enumerable.Repeat(0, N).Select(_ => NextInt).ToList();
        static List<long> NextLongList(int N) => Enumerable.Repeat(0, N).Select(_ => NextLong).ToList();
        static List<double> NextDoubleList(int N) => Enumerable.Repeat(0, N).Select(_ => NextDouble).ToList();
        static List<string> NextStringList(int N) => Enumerable.Repeat(0, N).Select(_ => NextString).ToList();
        static IEnumerable<T> OrderByRand<T>(this IEnumerable<T> x) => x.OrderBy(_ => rand.Next());
        static IEnumerable<T> Repeat<T>(T v, int n) => Enumerable.Repeat<T>(v, n);
        static List<T> Sort<T>(List<T> l) where T : IComparable
        {
            var tmp = l.ToArray(); Array.Sort(tmp); return tmp.ToList();
        }
        static List<T> Sort<T>(List<T> l, Comparison<T> comp) where T : IComparable
        {
            var tmp = l.ToArray(); Array.Sort(tmp, comp); return tmp.ToList();
        }
        static List<T> RevSort<T>(List<T> l) where T : IComparable
        {
            var tmp = l.ToArray(); Array.Sort(tmp, (x, y) => y.CompareTo(x)); return tmp.ToList();
        }
        static List<T> RevSort<T>(List<T> l, Comparison<T> comp) where T : IComparable
        {
            var tmp = l.ToArray(); Array.Sort(tmp, (x, y) => comp(y, x)); return tmp.ToList();
        }
        static IEnumerable<long> Prime(long x)
        {
            if (x < 2) yield break;
            yield return 2;
            var halfx = x / 2;
            var table = new bool[halfx + 1];
            var max = (long)(Math.Sqrt(x) / 2);
            for (long i = 1; i <= max; ++i)
            {
                if (table[i]) continue;
                var add = 2 * i + 1;
                yield return add;
                for (long j = 2 * i * (i + 1); j <= halfx; j += add) table[j] = true;
            }
            for (long i = max + 1; i <= halfx; ++i) if (!table[i] && 2 * i + 1 <= x) yield return 2 * i + 1;
        }
        static IEnumerable<long> Divisor(long x)
        {
            if (x < 1) yield break;
            var max = (long)Math.Sqrt(x);
            for (long i = 1; i <= max; ++i)
            {
                if (x % i != 0) continue;
                yield return i;
                if (i != x / i) yield return x / i;
            }
        }
        static long GCD(long a, long b)
        {
            while (b > 0) { var tmp = b; b = a % b; a = tmp; }
            return a;
        }
        class PQ<T> where T : IComparable
        {
            private List<T> h;
            private Comparison<T> c;
            public PQ(int cap, Comparison<T> c, bool asc = true) { h = new List<T>(cap); this.c = asc ? c : (x, y) => c(y, x); }
            public PQ(Comparison<T> c, bool asc = true) { h = new List<T>(); this.c = asc ? c : (x, y) => c(y, x); }
            public PQ(int cap, bool asc = true) : this(cap, (x, y) => x.CompareTo(y), asc) { }
            public PQ(bool asc = true) : this((x, y) => x.CompareTo(y), asc) { }
            public void Push(T v)
            {
                var i = h.Count;
                h.Add(v);
                while (i > 0)
                {
                    var ni = (i - 1) / 2;
                    if (c(v, h[ni]) >= 0) break;
                    h[i] = h[ni]; i = ni;
                }
                h[i] = v;
            }
            public T Peek => h[0];
            public int Count => h.Count;
            public T Pop()
            {
                var r = h[0];
                var v = h[h.Count - 1];
                h.RemoveAt(h.Count - 1);
                if (h.Count == 0) return r;
                var i = 0;
                while (i * 2 + 1 < h.Count)
                {
                    var i1 = i * 2 + 1;
                    var i2 = i * 2 + 2;
                    if (i2 < h.Count && c(h[i1], h[i2]) > 0) i1 = i2;
                    if (c(v, h[i1]) <= 0) break;
                    h[i] = h[i1]; i = i1;
                }
                h[i] = v;
                return r;
            }
        }
        class PQ<TKey, TValue> where TKey : IComparable
        {
            private PQ<Tuple<TKey, TValue>> q;
            public PQ(int cap, Comparison<TKey> c, bool asc = true) { q = new PQ<Tuple<TKey, TValue>>(cap, (x, y) => c(x.Item1, y.Item1), asc); }
            public PQ(Comparison<TKey> c, bool asc = true) { q = new PQ<Tuple<TKey, TValue>>((x, y) => c(x.Item1, y.Item1), asc); }
            public PQ(int cap, bool asc = true) : this(cap, (x, y) => x.CompareTo(y), asc) { }
            public PQ(bool asc = true) : this((x, y) => x.CompareTo(y), asc) { }
            public void Push(TKey k, TValue v) => q.Push(Tuple.Create(k, v));
            public Tuple<TKey, TValue> Peek => q.Peek;
            public int Count => q.Count;
            public Tuple<TKey, TValue> Pop() => q.Pop();
        }
        class Mod
        {
            static public long _mod = 1000000007;
            private long _val = 0;
            public Mod(long x) { if ((_val = x % _mod) < 0) _val += _mod; }
            static public implicit operator Mod(long x) => new Mod(x);
            static public explicit operator long(Mod x) => x._val;
            static public Mod operator +(Mod x) => x._val;
            static public Mod operator -(Mod x) => -x._val;
            static public Mod operator ++(Mod x) => x._val + 1;
            static public Mod operator --(Mod x) => x._val - 1;
            static public Mod operator +(Mod x, Mod y) => (x._val + y._val) % _mod;
            static public Mod operator -(Mod x, Mod y) => ((((x._val - y._val) % _mod) + _mod) % _mod);
            static public Mod operator *(Mod x, Mod y) => (x._val * y._val) % _mod;
            static public Mod operator /(Mod x, Mod y) => (x._val * Inverse(y._val)) % _mod;
            static public bool operator ==(Mod x, Mod y) => x._val == y._val;
            static public bool operator !=(Mod x, Mod y) => x._val != y._val;
            static public bool operator <(Mod x, Mod y) => x._val < y._val;
            static public bool operator >(Mod x, Mod y) => x._val > y._val;
            static public bool operator <=(Mod x, Mod y) => x._val <= y._val;
            static public bool operator >=(Mod x, Mod y) => x._val >= y._val;
            static public Mod Pow(Mod x, long y)
            {
                Mod a = 1;
                while (y != 0)
                {
                    if ((y & 1) == 1) a *= x;
                    x *= x;
                    y >>= 1;
                }
                return a;
            }
            static public long Inverse(long x)
            {
                long b = _mod, r = 1, u = 0, t = 0;
                while (b > 0)
                {
                    var q = x / b;
                    t = u; u = r - q * u; r = t;
                    t = b; b = x - q * b; x = t;
                }
                return r < 0 ? r + _mod : r;
            }
            override public bool Equals(object obj) => obj == null ? false : _val == ((Mod)obj)._val;
            public bool Equals(Mod obj) => obj == null ? false : _val == obj._val;
            public override int GetHashCode() => _val.GetHashCode();
            public override string ToString() => _val.ToString();
            static private List<Mod> _fact = new List<Mod>();
            static private List<Mod> _ifact = new List<Mod>();
            static private void Build(int n)
            {
                if (n >= _fact.Count)
                    for (int i = _fact.Count; i <= n; ++i)
                        if (i == 0L) { _fact.Add(1); _ifact.Add(1); }
                        else { _fact.Add(_fact[i - 1] * i); _ifact.Add(_ifact[i - 1] * Mod.Pow(i, _mod - 2)); }
            }
            static public Mod Comb(int n, int k)
            {
                Build(n);
                if (n == 0 && k == 0) return 1;
                if (n < k || n < 0) return 0;
                return _ifact[n - k] * _ifact[k] * _fact[n];
            }
        }
        class Mat<T>
        {
            private T[,] m;
            public Mat(T[,] v) { m = (T[,])v.Clone(); }
            static public implicit operator Mat<T>(T[,] v) => new Mat<T>(v);
            public T this[int r, int c] { get { return m[r, c]; } set { m[r, c] = value; } }
            static public Mat<T> operator +(Mat<T> a, T x)
            {
                var tm = (T[,])a.m.Clone();
                for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] += (dynamic)x;
                return new Mat<T>(tm);
            }
            static public Mat<T> operator +(Mat<T> a, Mat<T> b)
            {
                var tm = (T[,])a.m.Clone();
                for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] += (dynamic)b[r, c];
                return new Mat<T>(tm);
            }
            static public Mat<T> operator -(Mat<T> a, T x)
            {
                var tm = (T[,])a.m.Clone();
                for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] -= (dynamic)x;
                return new Mat<T>(tm);
            }
            static public Mat<T> operator -(Mat<T> a, Mat<T> b)
            {
                var tm = (T[,])a.m.Clone();
                for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] -= (dynamic)b[r, c];
                return new Mat<T>(tm);
            }
            static public Mat<T> operator *(Mat<T> a, T x)
            {
                var tm = (T[,])a.m.Clone();
                for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] *= (dynamic)x;
                return new Mat<T>(tm);
            }
            static public Mat<T> operator *(Mat<T> a, Mat<T> b)
            {
                var nr = a.m.GetLength(0);
                var nc = b.m.GetLength(1);
                var tm = new T[nr, nc];
                for (int i = 0; i < nr; ++i) for (int j = 0; j < nc; ++j) tm[i, j] = (dynamic)0;
                for (int r = 0; r < nr; ++r) for (int c = 0; c < nc; ++c) for (int i = 0; i < a.m.GetLength(1); ++i) tm[r, c] += a[r, i] * (dynamic)b[i, c];
                return new Mat<T>(tm);
            }
            static public Mat<T> Pow(Mat<T> x, long y)
            {
                var n = x.m.GetLength(0);
                var t = (Mat<T>)new T[n, n];
                for (int i = 0; i < n; ++i) for (int j = 0; j < n; ++j) t[i, j] = (dynamic)(i == j ? 1 : 0);
                while (y != 0)
                {
                    if ((y & 1) == 1) t *= x;
                    x *= x;
                    y >>= 1;
                }
                return t;
            }
        }
        static Mat<T> Pow<T>(Mat<T> x, long y) => Mat<T>.Pow(x, y);
        static Mod Pow(Mod x, long y) => Mod.Pow(x, y);
        static long Pow(long x, long y)
        {
            long a = 1;
            while (y != 0)
            {
                if ((y & 1) == 1) a *= x;
                x *= x;
                y >>= 1;
            }
            return a;
        }
        static List<long> _fact = new List<long>();
        static void _Build(int n)
        {
            if (n >= _fact.Count)
                for (int i = _fact.Count; i <= n; ++i)
                    if (i == 0L) _fact.Add(1);
                    else _fact.Add(_fact[i - 1] * i);
        }
        static long Comb(int n, int k)
        {
            _Build(n);
            if (n == 0 && k == 0) return 1;
            if (n < k || n < 0) return 0;
            return _fact[n] / _fact[k] / _fact[n - k];
        }
    }
}