New Year Contest 2015 I マージ
こういうの苦手
問題概要(原文)
$\{1, 2, \ldots , N \}$ を並び替えた順列 $P,Q$ が与えられる．以下のルールに従って，数列 $R$ を作る．作れる数列の通り数を $\text{MOD} \; 10^9 + 7$ で求めよ．
$N \times N$ の格子を $(0,\,0)$ から $(N,\,N)$ に移動する通り数のようにして考えよう． ある場所 $(i,\,j)$ から $(i+k,\,j+k)$ に移動したとき，移動の仕方が異なるにも関わらず，同じ数列ができてしまう場合を考える． 例えば $k=1$ として， $P[i] = Q[j]$ のとき条件を満たす． 同様にして， $P[i] = Q[j], P[i+1] = Q[j+1], \ldots, P[i+k] = Q[i+k]$ を満たすとき，条件を満たすような数列が存在する．
これを差し引いてやることを考えよう．先ほどの条件から $k \times k$ の格子を線対称に進むと同じ数列ができてしまうことが分かる．そうでないものを数え上げよう．
ソースコード
using System;
using System.Linq;
using System.Linq.Expressions;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using StringBuilder = System.Text.StringBuilder;
using System.Numerics;
using Number = System.Int64;
namespace Program
{
public class Solver
{
public void Solve()
{
const long MOD = (long)1e9 + 7;
var n = sc.Integer();
var a = sc.Integer(n);
var b = sc.Integer(n);
var dp = new long[n + 1, n + 1];
dp[0, 0] = 1;
var C = new long[n + 1];
C[0] = 1;
for (int i = 0; i < n; i++)
for (int j = 0; j <= i; j++)
C[i + 1] = (C[i + 1] + C[j] * C[i - j]) % MOD;
Debug.WriteLine(C.AsJoinedString());
for (int i = 0; i <= n; i++)
for (int j = 0; j <= n; j++)
{
var p = i - 1; var q = j - 1;
var l = 0;
while (0 <= p && 0 <= q && a[p] == b[q])
{
dp[i, j] -= (C[l] * dp[p, q]) % MOD;
dp[i, j] = (dp[i, j] + MOD) % MOD;
p--; q--; l++;
}
if (i < n) dp[i + 1, j] = (dp[i + 1, j] + dp[i, j]) % MOD;
if (j < n) dp[i, j + 1] = (dp[i, j + 1] + dp[i, j]) % MOD;
}
IO.Printer.Out.WriteLine(dp[n, 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
コメント
- なんとか解けたという感じ
