結果

問題 No.271 next_permutation (2)
ユーザー anta
提出日時 2015-06-28 20:11:54
言語 C++11（廃止可能性あり）
(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

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#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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