Problem:
Score : 600 points

Problem Statement

You are given a string S of length 2N consisting of lowercase English letters.

There are 2^{2N} ways to color each character in S red or blue. Among these ways, how many satisfy the following condition?

The string obtained by reading the characters painted red from left to right is equal to the string obtained by reading the characters painted blue from right to left.

Constraints

1 \leq N \leq 18

The length of S is 2N.

S consists of lowercase English letters.

Input

Input is given from Standard Input in the following format:

N
S

Output

Print the number of ways to paint the string that satisfy the condition.

Sample Input 1

4
cabaacba

Sample Output 1

4

There are four ways to paint the string, as follows:

cabaacba

cabaacba

cabaacba

cabaacba

Sample Input 2

11
mippiisssisssiipsspiim

Sample Output 2

504

Sample Input 3

4
abcdefgh

Sample Output 3

0

Sample Input 4

18
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

Sample Output 4

9075135300

The answer may not be representable as a 32-bit integer.

Solution in C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using static System.Math;
using System.Text;
using System.Threading;
using System.Globalization;
using System.Runtime.CompilerServices;
using Library;

namespace Program
{
    public static class ProblemA
    {
        static bool SAIKI = false;
        static public int numberOfRandomCases = 0;
        static public void MakeTestCase(List<string> _input, List<string> _output, ref Func<string[], bool> _outputChecker)
        {
        }
        static public void Solve()
        {
            var N = NN;
            var S = NS;
            var str = new[] { S.Substring(0, (int)N).ToArray(), S.Substring((int)N).Reverse().ToArray() };
            var counter = new[] { new LIB_Dict<string, long>(), new LIB_Dict<string, long>() };
            var ten = LIB_Math.Pow(2, N);
            for (var t = 0; t < 2; t++)
                for (var i = 0; i < ten; i++)
                {
                    var ary = new List<long>();
                    var ary2 = new List<long>();
                    for (var j = 0; j < N; j++) (LIB_BitUtil.IsSet(i, j) ? ary : ary2).Add(str[t][j]);
                    counter[t][ary.Join() + "," + ary2.Join()]++;
                }
            Console.WriteLine(counter[0].Sum(e => e.Value * counter[1][e.Key]));
        }
        class Printer : StreamWriter
        {
            public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
            public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { base.AutoFlush = false; }
            public Printer(Stream stream, Encoding encoding) : base(stream, encoding) { base.AutoFlush = false; }
        }
        static LIB_FastIO fastio = new LIB_FastIODebug();
        static public void Main(string[] args) { if (args.Length == 0) { fastio = new LIB_FastIO(); Console.SetOut(new Printer(Console.OpenStandardOutput())); } if (SAIKI) { var t = new Thread(Solve, 134217728); t.Start(); t.Join(); } else Solve(); Console.Out.Flush(); }
        static long NN => fastio.Long();
        static double ND => fastio.Double();
        static string NS => fastio.Scan();
        static long[] NNList(long N) => Repeat(0, N).Select(_ => NN).ToArray();
        static double[] NDList(long N) => Repeat(0, N).Select(_ => ND).ToArray();
        static string[] NSList(long N) => Repeat(0, N).Select(_ => NS).ToArray();
        static long Count<T>(this IEnumerable<T> x, Func<T, bool> pred) => Enumerable.Count(x, pred);
        static IEnumerable<T> Repeat<T>(T v, long n) => Enumerable.Repeat<T>(v, (int)n);
        static IEnumerable<int> Range(long s, long c) => Enumerable.Range((int)s, (int)c);
        static IOrderedEnumerable<T> OrderByRand<T>(this IEnumerable<T> x) => Enumerable.OrderBy(x, _ => xorshift);
        static IOrderedEnumerable<T> OrderBy<T>(this IEnumerable<T> x) => Enumerable.OrderBy(x.OrderByRand(), e => e);
        static IOrderedEnumerable<T1> OrderBy<T1, T2>(this IEnumerable<T1> x, Func<T1, T2> selector) => Enumerable.OrderBy(x.OrderByRand(), selector);
        static IOrderedEnumerable<T> OrderByDescending<T>(this IEnumerable<T> x) => Enumerable.OrderByDescending(x.OrderByRand(), e => e);
        static IOrderedEnumerable<T1> OrderByDescending<T1, T2>(this IEnumerable<T1> x, Func<T1, T2> selector) => Enumerable.OrderByDescending(x.OrderByRand(), selector);
        static IOrderedEnumerable<string> OrderBy(this IEnumerable<string> x) => x.OrderByRand().OrderBy(e => e, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<T> OrderBy<T>(this IEnumerable<T> x, Func<T, string> selector) => x.OrderByRand().OrderBy(selector, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<string> OrderByDescending(this IEnumerable<string> x) => x.OrderByRand().OrderByDescending(e => e, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<T> OrderByDescending<T>(this IEnumerable<T> x, Func<T, string> selector) => x.OrderByRand().OrderByDescending(selector, StringComparer.OrdinalIgnoreCase);
        static string Join<T>(this IEnumerable<T> x, string separator = "") => string.Join(separator, x);
        static uint xorshift { get { _xsi.MoveNext(); return _xsi.Current; } }
        static IEnumerator<uint> _xsi = _xsc();
        static IEnumerator<uint> _xsc() { uint x = 123456789, y = 362436069, z = 521288629, w = (uint)(DateTime.Now.Ticks & 0xffffffff); while (true) { var t = x ^ (x << 11); x = y; y = z; z = w; w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)); yield return w; } }
    }
}
namespace Library {
    class LIB_Dict<K, V> : Dictionary<K, V>
    {
        Func<K, V> d;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_Dict(Func<K, V> _d) { d = _d; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_Dict() : this(_ => default(V)) { }
        new public V this[K i]
        {
            get
            {
                V v;
                return TryGetValue(i, out v) ? v : base[i] = d(i);
            }
            set
            {
                base[i] = value;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override int GetHashCode()
        {
            unchecked
            {
                var ret = 23;
                for (var e = GetEnumerator(); e.MoveNext();) ret = (ret * 37) ^ (e.Current.Key.GetHashCode() << 5 + e.Current.Key.GetHashCode()) ^ e.Current.Value.GetHashCode();
                return ret;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override bool Equals(object obj)
        {
            var o = (LIB_Dict<K, V>)obj;
            foreach (var kv in o)
            {
                if (!ContainsKey(kv.Key) || !this[kv.Key].Equals(kv.Value))
                    return false;
            }
            return o.Count == Count;
        }
    }
    class LIB_Math
    {
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<long> Primes(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;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<long> Factors(long x)
        {
            if (x < 2) yield break;
            while (x % 2 == 0)
            {
                x /= 2; yield return 2;
            }
            var max = (long)Math.Sqrt(x);
            for (long i = 3; i <= max; i += 2)
            {
                while (x % i == 0)
                {
                    x /= i; yield return i;
                }
            }
            if (x != 1) yield return x;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public 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;
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long GCD(long a, long b)
        {
            while (b > 0)
            {
                var tmp = b;
                b = a % b;
                a = tmp;
            }
            return a;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long LCM(long a, long b) => a / GCD(a, b) * b;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long Pow(long x, long y)
        {
            long a = 1;
            while (y != 0)
            {
                if ((y & 1) == 1) a *= x;
                if (x < long.MaxValue / x) x *= x;
                y >>= 1;
            }
            return a;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public decimal Sqrt(decimal x)
        {
            decimal prev, cur = (decimal)Math.Sqrt((double)x);
            do
            {
                prev = cur;
                if (prev == 0) return 0;
                cur = (prev + x / prev) / 2;
            } while (cur != prev);
            return cur;
        }
        static List<long> _fact = new List<long>() { 1 };
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static void Build(long n)
        {
            if (n >= _fact.Count)
                for (int i = _fact.Count; i <= n; ++i) _fact.Add(_fact[i - 1] * i);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long Comb(long n, long k)
        {
            Build(n);
            if (n == 0 && k == 0) return 1;
            if (n < k || n < 0) return 0;
            return _fact[(int)n] / _fact[(int)(n - k)] / _fact[(int)k];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long Perm(long n, long k)
        {
            Build(n);
            if (n == 0 && k == 0) return 1;
            if (n < k || n < 0) return 0;
            return _fact[(int)n] / _fact[(int)(n - k)];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<List<int>> MakePermutation(long n, bool zeroIndexed = true)
        {
            if (n <= 0) throw new Exception();
            var c = new int[n];
            var a = new int[n];
            if (!zeroIndexed) a[0] = 1;
            for (var i = 1; i < n; i++) a[i] = a[i - 1] + 1;
            yield return new List<int>(a);
            for (var i = 0; i < n;)
            {
                if (c[i] < i)
                {
                    if (i % 2 == 0)
                    {
                        var t = a[0]; a[0] = a[i]; a[i] = t;
                    }
                    else
                    {
                        var t = a[c[i]]; a[c[i]] = a[i]; a[i] = t;
                    }
                    yield return new List<int>(a);
                    ++c[i];
                    i = 0;
                }
                else
                {
                    c[i] = 0;
                    ++i;
                }
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public bool NextPermutation(long[] ary)
        {
            var n = ary.Length;
            var i = n - 1;
            while (i - 1 >= 0 && ary[i - 1] > ary[i]) --i;
            if (i == 0) return false;
            var j = i;
            while (j + 1 < n && ary[i - 1] < ary[j + 1]) ++j;
            var tmp = ary[i - 1]; ary[i - 1] = ary[j]; ary[j] = tmp;
            var s = i;
            var t = n - 1;
            while (t - s > 0)
            {
                tmp = ary[t]; ary[t] = ary[s]; ary[s] = tmp;
                ++s; --t;
            }
            return true;
        }
    }
    class LIB_BitUtil
    {
        public static readonly ulong[] BitMask;
        static LIB_BitUtil()
        {
            BitMask = new ulong[64];
            BitMask[0] = 1;
            for (var i = 1; i < 64; i++) BitMask[i] = BitMask[i - 1] << 1;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public IEnumerable<long> ScanOne(long value)
        {
            for (; value > 0; value &= value - 1) yield return value & -value;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long PopCount(ulong value)
        {
            value = (value & 0x5555555555555555) + ((value >> 1) & 0x5555555555555555);
            value = (value & 0x3333333333333333) + ((value >> 2) & 0x3333333333333333);
            value = (value & 0x0f0f0f0f0f0f0f0f) + ((value >> 4) & 0x0f0f0f0f0f0f0f0f);
            value = (value & 0x00ff00ff00ff00ff) + ((value >> 8) & 0x00ff00ff00ff00ff);
            value = (value & 0x0000ffff0000ffff) + ((value >> 16) & 0x0000ffff0000ffff);
            value = (value & 0x00000000ffffffff) + ((value >> 32) & 0x00000000ffffffff);
            return (long)value;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public long PopCount(long value) => PopCount((ulong)value);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public bool IsSet(ulong value, int idx) => (value & BitMask[idx]) != 0;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static public bool IsSet(long value, int idx) => IsSet((ulong)value, idx);
    }
    class LIB_FastIO
    {
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_FastIO() { str = Console.OpenStandardInput(); }
        readonly Stream str;
        readonly byte[] buf = new byte[1024];
        int len, ptr;
        public bool isEof = false;
        public bool IsEndOfStream
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get { return isEof; }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        byte read()
        {
            if (isEof) throw new EndOfStreamException();
            if (ptr >= len)
            {
                ptr = 0;
                if ((len = str.Read(buf, 0, 1024)) <= 0)
                {
                    isEof = true;
                    return 0;
                }
            }
            return buf[ptr++];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        char Char()
        {
            byte b = 0;
            do b = read();
            while (b < 33 || 126 < b);
            return (char)b;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public long Long()
        {
            long ret = 0; byte b = 0; var ng = false;
            do b = read();
            while (b != '-' && (b < '0' || '9' < b));
            if (b == '-') { ng = true; b = read(); }
            for (; true; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                else ret = ret * 10 + b - '0';
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public double Double() { return double.Parse(Scan(), CultureInfo.InvariantCulture); }
    }
    class LIB_FastIODebug : LIB_FastIO
    {
        Queue<string> param = new Queue<string>();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        string NextString() { if (param.Count == 0) foreach (var item in Console.ReadLine().Split(' ')) param.Enqueue(item); return param.Dequeue(); }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_FastIODebug() { }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override string Scan() => NextString();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override long Long() => long.Parse(NextString());
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override double Double() => double.Parse(NextString());
    }
}