結果

問題 No.271 next_permutation (2)
ユーザー antaanta
提出日時 2015-06-28 20:11:54
言語 C++11
(gcc 13.3.0)
結果
AC  
実行時間 23 ms / 2,000 ms
コード長 6,489 bytes
コンパイル時間 934 ms
コンパイル使用メモリ 94,192 KB
実行使用メモリ 6,820 KB
最終ジャッジ日時 2024-11-26 07:45:50
合計ジャッジ時間 2,235 ms
ジャッジサーバーID
（参考情報） 		judge3 / judge1
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ファイルパターン 結果
other AC * 21
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ソースコード

diff # 

#undef NDEBUG
#include <string>
#include <vector>
#include <algorithm>
#include <numeric>
#include <set>
#include <map>
#include <queue>
#include <iostream>
#include <sstream>
#include <cstdio>
#include <cmath>
#include <ctime>
#include <cstring>
#include <cctype>
#include <cassert>
#include <limits>
#include <functional>
#define rep(i,n) for(int (i)=0;(i)<(int)(n);++(i))
#define rer(i,l,u) for(int (i)=(int)(l);(i)<=(int)(u);++(i))
#define reu(i,l,u) for(int (i)=(int)(l);(i)<(int)(u);++(i))
#if defined(_MSC_VER) || __cplusplus > 199711L
#define aut(r,v) auto r = (v)
#else
#define aut(r,v) __typeof(v) r = (v)
#endif
#define each(it,o) for(aut(it, (o).begin()); it != (o).end(); ++ it)
#define all(o) (o).begin(), (o).end()
#define pb(x) push_back(x)
#define mp(x,y) make_pair((x),(y))
#define mset(m,v) memset(m,v,sizeof(m))
#define INF 0x3f3f3f3f
#define INFL 0x3f3f3f3f3f3f3f3fLL
using namespace std;
typedef vector<int> vi; typedef pair<int,int> pii; typedef vector<pair<int,int> > vpii; typedef long long ll;
template<typename T, typename U> inline void amin(T &x, U y) { if(y < x) x = y; }
template<typename T, typename U> inline void amax(T &x, U y) { if(x < y) x = y; }


#ifdef NDEBUG
#error assert is disabled!
#endif

template<typename T>
T readNatural(T lo, T up) {
	assert(0 <= lo && lo <= up);
	T x = 0;
	while(1) {
		int d = getchar();
		if(!('0' <= d && d <= '9')) {
			ungetc(d, stdin);
			break;
		}
		d -= '0';
		assert(d <= up && x <= (up - d) / 10);
		x = x * 10 + d;
	}
	assert(lo <= x && x <= up);
	return x;
}

void readSpace() { int c = getchar(); assert(c == ' '); }
static bool read_eof = false;
void readEOL() {
	int c = getchar();
	if(c == EOF) {
		assert(!read_eof);
		read_eof = true;
	}else {
		assert(c == '\n');
	}
}


template<int MOD>
struct ModInt {
	static const int Mod = MOD;
	unsigned x;
	ModInt(): x(0) { }
	ModInt(signed sig) { int sigt = sig % MOD; if(sigt < 0) sigt += MOD; x = sigt; }
	ModInt(signed long long sig) { int sigt = sig % MOD; if(sigt < 0) sigt += MOD; x = sigt; }
	int get() const { return (int)x; }
	
	ModInt &operator+=(ModInt that) { if((x += that.x) >= MOD) x -= MOD; return *this; }
	ModInt &operator-=(ModInt that) { if((x += MOD - that.x) >= MOD) x -= MOD; return *this; }
	ModInt &operator*=(ModInt that) { x = (unsigned long long)x * that.x % MOD; return *this; }
	ModInt &operator/=(ModInt that) { return *this *= that.inverse(); }
	
	ModInt operator+(ModInt that) const { return ModInt(*this) += that; }
	ModInt operator-(ModInt that) const { return ModInt(*this) -= that; }
	ModInt operator*(ModInt that) const { return ModInt(*this) *= that; }
	ModInt operator/(ModInt that) const { return ModInt(*this) /= that; }
	
	ModInt inverse() const {
		signed a = x, b = MOD, u = 1, v = 0;
		while(b) {
			signed t = a / b;
			a -= t * b; std::swap(a, b);
			u -= t * v; std::swap(u, v);
		}
		if(u < 0) u += Mod;
		ModInt res; res.x = (unsigned)u;
		return res;
	}
};
typedef ModInt<1000000007> mint;


struct FenwickTree {
	typedef int T;
	vector<T> v;
	void init(int n) { v.assign(n, 0); }
	void add(int i, T x) {
		for(; i < (int)v.size(); i |= i+1) v[i] += x;
	}
	T sum(int i) const {	//[0, i)
		T r = 0;
		for(-- i; i >= 0; i = (i & (i+1)) - 1) r += v[i];
		return r;
	}
	T sum(int left, int right) const {	//[left, right)
		return sum(right) - sum(left);
	}
};

//0 <= res[i] < n - i
void permutationToFactorialNumberSystem(const vector<int> &p, vector<int> &res) {
	int n = p.size();
	FenwickTree ft;
	ft.init(n);
	res.resize(n);
	for(int i = n - 1; i >= 0; -- i) {
		res[i] = ft.sum(p[i]);
		ft.add(p[i], 1);
	}
}
long long integerToFactorialNumberSystem(int n, long long x, vector<int> &res) {
	assert(x >= 0);
	res.resize(n);
	for(int i = n - 1; i >= 0; -- i) {
		res[i] = x % (n - i);
		x /= n - i;
	}
	return x;
}
int addFactorialNumberSystem(const vector<int> &x, const vector<int> &y, vector<int> &res) {
	int n = x.size();
	assert(y.size() == n);
	res.resize(n);
	int carry = 0;
	for(int i = n - 1; i >= 0; -- i) {
		int z = x[i] + y[i] + carry;
		res[i] = z % (n - i);
		carry = z / (n - i);
	}
	return carry;
}

//sum [l,u)
ll arithsum(int l, int u) {
	if(l >= u) return 0;
	int w = u - l;
	return (ll)l * w + (ll)w * (w-1) / 2;
}

struct DP {
	vector<pair<mint,mint> > dp;
	DP(int n): dp((n+1) * 2) { }
	pair<mint,mint> &operator()(int i, bool lt) { return dp[i * 2 + lt]; }
};

//  p < x となる置換のinvの和
//= p < x となるfactorial number systemの桁の和の和
//(i, lt)に対して(cnt, sum)を持つ桁DPをする
mint solve(const vector<int> &x) {
	int n = x.size();
	DP dp(n);
	dp(n, true) = mp(mint(1), mint(0));
	for(int i = n-1; i >= 0; -- i) rep(lt, 2) {
		mint cnt, sum;
		int d = x[i];
		/*	//以下と同じことをまとめて数えてやる
		rep(e, n - i) if(lt || e <= d) {
			auto p = dp(i+1, lt || e < d);
			cnt += p.first, sum += p.second;
			sum += p.first * e;
		}
		*/
		//e < d の場合
		{
			auto p = dp(i+1, true);
			cnt += p.first * d, sum += p.second * d;
			sum += p.first * arithsum(0, d);
		}
		//e = d の場合
		{
			auto p = dp(i+1, lt != 0);
			cnt += p.first, sum += p.second;
			sum += p.first * d;
		}
		//e > d の場合
		if(lt) {
			auto p = dp(i+1, true);
			int t = n - i - d - 1;
			cnt += p.first * t, sum += p.second * t;
			sum += p.first * arithsum(d+1, n - i);
		}
		dp(i, lt != 0) = mp(cnt, sum);
	}
	return dp(0, false).second;
}

int main() {
	int N; long long K;
	N = readNatural(1, (int)1e5);
	readSpace();
	K = readNatural(0LL, (ll)1e18);
	readEOL();
	vector<int> p(N);
	rep(i, N) {
		p[i] = readNatural(1, N);
		if(i < N-1) readSpace(); else readEOL();
	}
	assert(!read_eof);
	{	vector<bool> vis(N);
		rep(i, N) {
			assert(!vis[p[i]-1]);
			vis[p[i]-1] = true;
		}
	}
	rep(i, N) -- p[i];

	//p, f^K(p) を factorial number system に変換する
	//キャリーは覚えておいて、それは別に足す
	vector<int> x, y, z;
	permutationToFactorialNumberSystem(p, x);
	long long cycles = 0;
	cycles += integerToFactorialNumberSystem(N, K, y);
	cycles += addFactorialNumberSystem(x, y, z);

	//factorial number systemでの要素ごとのinvへの付与を考えると、
	//  \sum_{p is a permutation of [1..n]} inv(p)
	//= \sum_{i=2}^n \sum{j=0}^{i-1} (n! / i) j
	//= n! \sum_{i=1}^{n-1} i / 2
	//= n! n (n-1) / 4
	mint allinvsum = 1;
	rep(i, N) allinvsum *= i+1;
	allinvsum *= mint(N) * (N-1) / 4;

	mint ans;
	ans += allinvsum * cycles;
	ans += solve(z);
	ans -= solve(x);

	printf("%d\n", ans.get());
	return 0;
}
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