AGC002F Leftmost Ball

問題概要(原文)

$1,2,3, \ldots ,N$ という $N$ 色のボールが $K$ 個ずつある.これを一列に並べる. その後,各色のボールのうち,一番左側にあるボールを色 $0$ に塗る. 色の並びは何通りあるか $\mathrm{mod} 10^{9} + 7$ で求めよ.

考察

$K = 1$ ならば $1$ 通り.$K \geq 1$ とする.

最終的な並びにおいて,色 $1,2,3,\ldots, N$ の順番で出現したことにする. このとき,色の並びは何通りあるか?という問題は組合せをうまくやると解ける.

しかし,今回は色 $0$ があってややこしい. 色 $0$ と,各色の順番の関係について考えよう. まず色 $1,2,3, \ldots, N$ の順番で出現する. 色 $i$ が出現するのは $i$ 番目の $0$ が出現した以降である.

つまり,こんなグラフになる.

このグラフでの並びの通り数を求めればよく,状態としては

  • $0$ 番目が何回出現したか
  • 何番目の色まで並べ終わったか

の $2$ つで $O(N^{2})$

遷移は以下の $2$ 通り

  • 色 $0$ のボールを今置ける左端に置く
  • 色 $i$ のボールを今置ける左端に $1$ つ, $K-2$ 個を残った場所に適当に置く

後者は置ける位置を $X$ 個として $\binom{X}{K-2}$ となる. これは前もって階乗を求めておけば $O(1)$ でできる.

全体として $O((N+K)N)$

ソースコード

using System;
using System.Linq;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using StringBuilder = System.Text.StringBuilder;
using System.Numerics;
using Point = System.Numerics.Complex;
using Number = System.Int64;
using C = System.Int32;
namespace Program
{
    public class Solver
    {
        public void Solve()
        {
            var n = sc.Integer();
            var m = sc.Integer();
            if (m == 1)
            {
                IO.Printer.Out.WriteLine(1);
                return;
            }
            var table = new ModTable(2001 * 2001);
            var dp = Enumerate(n + 2, x => new ModInteger[n + 2]);
            dp[0][0] = 1;
            for (int i = 0; i <= n; i++)
                for (int j = 0; j <= n; j++)
                {
                    var rem = n - i + (m - 1) * (n - j);

                    dp[i + 1][j] += dp[i][j];
                    if (j + 1 <= i) dp[i][j + 1] += dp[i][j] * table.Combination(rem - 1, (m - 2));
                }

            IO.Printer.Out.WriteLine(dp[n][n] * table.perm[n]);
        }
        public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput());
        static T[] Enumerate<T>(int n, Func<int, T> f) { var a = new T[n]; for (int i = 0; i < n; ++i) a[i] = f(i); return a; }
        static public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }
    }
}

#region main
static class Ex
{
    static public string AsString(this IEnumerable<char> ie) { return new string(System.Linq.Enumerable.ToArray(ie)); }
    static public string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ") { return string.Join(st, ie); }
    static public void Main()
    {
        var solver = new Program.Solver();
        solver.Solve();
        Program.IO.Printer.Out.Flush();
    }
}
#endregion
#region Ex
namespace Program.IO
{
    using System.IO;
    using System.Text;
    using System.Globalization;
    public class Printer: StreamWriter
    {
        static Printer() { Out = new Printer(Console.OpenStandardOutput()) { AutoFlush = false }; }
        public static Printer Out { get; set; }
        public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
        public Printer(System.IO.Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
        public Printer(System.IO.Stream stream, Encoding encoding) : base(stream, encoding) { }
        public void Write<T>(string format, T[] source) { base.Write(format, source.OfType<object>().ToArray()); }
        public void WriteLine<T>(string format, T[] source) { base.WriteLine(format, source.OfType<object>().ToArray()); }
    }
    public class StreamScanner
    {
        public StreamScanner(Stream stream) { str = stream; }
        public readonly Stream str;
        private readonly byte[] buf = new byte[1024];
        private int len, ptr;
        public bool isEof = false;
        public bool IsEndOfStream { get { return isEof; } }
        private byte read()
        {
            if (isEof) return 0;
            if (ptr >= len) { ptr = 0; if ((len = str.Read(buf, 0, 1024)) <= 0) { isEof = true; return 0; } }
            return buf[ptr++];
        }
        public char Char() { byte b = 0; do b = read(); while ((b < 33 || 126 < b) && !isEof); return (char)b; }

        public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }
        public string ScanLine()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b != '\n'; b = (char)read())
                if (b == 0) break;
                else if (b != '\r') sb.Append(b);
            return sb.ToString();
        }
        public long Long()
        {
            if (isEof) return long.MinValue;
            long ret = 0; byte b = 0; var ng = false;
            do b = read();
            while (b != 0 && b != '-' && (b < '0' || '9' < b));
            if (b == 0) return long.MinValue;
            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';
            }
        }
        public int Integer() { return (isEof) ? int.MinValue : (int)Long(); }
        public double Double() { var s = Scan(); return s != "" ? double.Parse(s, CultureInfo.InvariantCulture) : double.NaN; }
        private T[] enumerate<T>(int n, Func<T> f)
        {
            var a = new T[n];
            for (int i = 0; i < n; ++i) a[i] = f();
            return a;
        }

        public char[] Char(int n) { return enumerate(n, Char); }
        public string[] Scan(int n) { return enumerate(n, Scan); }
        public double[] Double(int n) { return enumerate(n, Double); }
        public int[] Integer(int n) { return enumerate(n, Integer); }
        public long[] Long(int n) { return enumerate(n, Long); }
    }
}
#endregion
#region ModNumber
public partial struct ModInteger
{
    public const long Mod = (long)1e9 + 7;
    public long num;
    public ModInteger(long n) : this() { num = n % Mod; if (num < 0) num += Mod; }
    public override string ToString() { return num.ToString(); }
    public static ModInteger operator +(ModInteger l, ModInteger r) { var n = l.num + r.num; if (n >= Mod) n -= Mod; return new ModInteger() { num = n }; }
    public static ModInteger operator -(ModInteger l, ModInteger r) { var n = l.num + Mod - r.num; if (n >= Mod) n -= Mod; return new ModInteger() { num = n }; }
    public static ModInteger operator *(ModInteger l, ModInteger r) { return new ModInteger(l.num * r.num); }
    public static ModInteger operator ^(ModInteger l, long r) { return ModInteger.Pow(l, r); }
    public static implicit operator ModInteger(long n) { return new ModInteger() { num = n }; }
    public static ModInteger Pow(ModInteger v, long k)
    {
        ModInteger ret = 1;
        var n = k;
        for (; n > 0; n >>= 1, v *= v)
        {
            if ((n & 1) == 1)
                ret = ret * v;
        }
        return ret;
    }
}
#endregion
#region Inverse
public partial struct ModInteger
{
    static public ModInteger Inverse(ModInteger v)
    {
        long p, q;
        ExGCD(v.num, Mod, out p, out q);
        return new ModInteger(p % Mod + Mod);
    }
    static public long ExGCD(long a, long b, out long x, out long y)
    {
        var u = new long[] { a, 1, 0 };
        var v = new long[] { b, 0, 1 };
        while (v[0] != 0)
        {
            var t = u[0] / v[0];
            for (int i = 0; i < 3; i++)
            {
                var tmp = u[i] - t * v[i];
                u[i] = v[i];
                v[i] = tmp;
            }
        }
        x = u[1];
        y = u[2];
        if (u[0] > 0)
            return u[0];
        for (int i = 0; i < 3; i++)
            u[i] = -u[i];
        return u[0];
    }
}
#endregion
#region ModTable
public class ModTable
{
    public ModInteger[] perm, invp;
    public ModTable(int n)
    {
        perm = new ModInteger[n + 1];
        invp = new ModInteger[n + 1];
        perm[0] = 1;
        for (int i = 1; i <= n; i++)
            perm[i] = perm[i - 1] * i;
        invp[n] = ModInteger.Inverse(perm[n]);
        for (int i = n - 1; i >= 0; i--)
            invp[i] = invp[i + 1] * (i + 1);
        invp[0] = invp[1];
    }
    public ModInteger Inverse(int k) { return invp[k]; }
    public ModInteger Permutation(int n, int k)
    {
        if (n < 0 || n >= perm.Length)
            return 0;
        if (k < 0 || k >= n)
            return 0;
        return perm[n] * invp[n - k];
    }
    public ModInteger Combination(int n, int r)
    {
        if (n < 0 || n >= perm.Length || r < 0 || r > n) return 0;
        return perm[n] * invp[n - r] * invp[r];
    }
    public ModInteger RepeatedCombination(int n, int k)
    {
        if (k == 0) return 1;
        return Combination(n + k - 1, k);
    }
}
#endregion

コメント

  • グラフを書いてみたけど,大変…
  • 解法が分かるとコードは異常に短い