yukicoder No.243 出席番号(2)
解き直し
問題概要(原文)
$(1,2, \ldots , N)$ の並び替えであるような数列 $p$ を考える．
$p_i \neq a_i$ を満たすような $p$ はいくつあるか？ $\mathrm{mod} \; 10^{9} +7$ で求めよ．
考察
数 $i$ で違反する，ということは全て独立である． そこで $dp(i)= i$ 個違反しているような $p$ の通り数，とする．
これを直接求めるのは難しい． まず $i$ 個違反するような選び方，というのを単純なDPで求める． その後 $(N-i)!$ をかけてやると，少なくとも $i$ 個違反している通り数になる．$i+1$ 個違反しているやつとかは全て求まっているのでここから上手く差し引けばよい．
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;
namespace Program
{
public class Solver
{
public void Solve()
{
var n = sc.Integer();
var a = new int[5000];
foreach (var x in sc.Integer(n)) a[x]++;
var dp = new ModInteger[n + 2];
dp[0] = 1;
for (int i = 0; i < n; i++)
if (a[i] > 0)
for (int j = n; j >= 0; j--)
if (dp[j].num != 0)
dp[j + 1] += dp[j] * a[i];
//dp[i] = i ko dame
var fact = new ModInteger[n + 1]; fact[0] = 1;
for (int i = 1; i <= n; i++)
fact[i] = fact[i - 1] * i;
var inv = Enumerate(n + 1, x => ModInteger.Pow(fact[x], ModInteger.Mod - 2));
for (int i = n; i >= 0; i--)
{
dp[i] *= fact[n - i];
for (int j = i + 1; j <= n; j++)
dp[i] -= fact[j] * inv[i] * inv[j - i] * dp[j];
}
IO.Printer.Out.WriteLine(dp[0]);
}
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
コメント
- ~K Perm Countingとほとんど同じになった…
