CodeForces 749E: Inversions After Shuffle

Posted 2022-07-11 pinkrabbit

题目传送门：CF749E。

记一道傻逼计数题。

题意简述：

给一个 \\(1\\) 到 \\(n\\) 的排列，随机选取区间 \\([l,r]\\) 随机打乱区间内的元素，问打乱后的整个序列的逆序数期望。

题解：

下面是代码，复杂度 \\(\\mathcalO(n\\log n)\\)。

#include <cstdio>
#include <cstring>

typedef long long LL;
typedef double db;
const int MN = 100005;

int N, A[MN];
LL pre[MN], suf[MN], Inv;
LL b0[MN], b1[MN], b2[MN];
inline void add(LL *b, int i, LL x)  for (; i <= N; i += i & -i) b[i] += x; 
inline LL qur(LL *b, int i)  LL a = 0; for (; i; i -= i & -i) a += b[i]; return a; 
db Ans;

int main() 
    scanf("%d", &N), Ans = (db)(N - 1) * (N + 2) / 24;
    for (int i = 1; i <= N; ++i) scanf("%d", &A[i]);
    for (int i = 1; i <= N; ++i) add(b0, A[i], 1), pre[i] = pre[i - 1] + i - qur(b0, A[i]);
    Inv = pre[N], memset(b0, 0, sizeof b0);
    for (int i = N; i >= 1; --i) suf[i] = suf[i + 1] + qur(b0, A[i]), add(b0, A[i], 1);
    Ans += 2 * Inv;
    for (int i = 1; i <= N; ++i) Ans -= 2. * (i * pre[i] + (N - i + 1) * suf[i]) / N / (N + 1);
    memset(b0, 0, sizeof b0);
    for (int i = N; i >= 1; --i) 
        Ans -= (db)(qur(b2, A[i]) * i * i + qur(b1, A[i]) * i + qur(b0, A[i])) / N / (N + 1);
        add(b0, A[i], (LL)i * (i - 1));
        add(b1, A[i], 1 - 2 * i);
        add(b2, A[i], 1);
    
    printf("%.15lf\\n", Ans);
    return 0;