POIXV Permutation

Description

Multiset is a mathematical object similar to a set, but each member of a multiset may have more than one membership. Just as with any set, the members of a multiset can be ordered in many ways. We call each such ordering a permutation of the multiset. For example, among the permutations of the multiset\((1,1,2,3,3,3,7,8)\) there are\((2,3,1,3,3,7,1,8)\) and\((8,7,3,3,3,2,1,1)\) .

We will say that one permutation of a given multiset is smaller (in lexicographic order) than another permutation, if on the first position that does not match the first permutation has a smaller element than the other one. All permutations of a given multiset can be numbered (starting from one) in an increasing order.

Write a programme that

• reads the description of a permutation of a multiset and a positive integer \(m\) from the standard input,
• determines the remainder of the rank of that permutation in the lexicographic ordering modulo \(m\),
• writes out the result to the standard output.

Input

The first line of the standard input holds two integers \(N\) and \(M\) \((1 \le N \le 3 \times 10^5, 2 \le m \le 10^9)\), separated by a single space. These denote, respectively, the cardinality of the multiset and the number \(m\). The second line of the standard input contains \(n\) positive integers \(a_i\) \((1 \le a_i \le 3 \times 10^5)\), separated by single spaces and denoting successive elements of the multiset permutation.

Output

The first and only line of the standard output is to hold one integer, the remainder modulo of the rank of the input permutation in the lexicographic ordering.

Sample Input

4 1000

2 1 10 2

Sample Output

5

首先我们考虑没有重复的元素的排列，则其序数（从\(0\)开始）为

\[X = \sum_{i = 1}^{N}(r_i-1)(N-i)!
\]

其中\(r_i\)表示\(a_i\)在未在排列前\(i-1\)位出现的元素的排名。这就是康托展开，用树状数组可以将复杂度优化到\(O(nlogn)\)。

下面我们考虑有重集合，用康托展开一样的方式思考。我们可以得出

\[X = \sum_{i = 1}^N \sum_{j < a_i}f_{i,j}
\]

其中\(j\)表示除去排列前\(i-1\)位的元素还剩的元素，\(f_{i,j}\)表示确定了前\(i-1\)位后第\(i\)位放\(j\)的所有可能的排列个数。用可重排列公式，不难得出

\[f_{i,j} = \frac{(N-i)!}{(\prod_{k = 1}^{j-1}c_k)(c_j-1))(\prod_{k = j+1}^{3 \times 10^5}c_k)}
\]

\(c_k\)表示确定了前\(i-1\)位后可重集合中\(k\)这个数的个数。

那么问题就来了，我们怎么去计算这个式子呢？

其实还是可以用树状数组来维护的。

首先有$$f_{i,j} = c_j\frac{(N-i)!}{\prod_{k = 1}^{3 \times 10^5}c_k}$$

所以

\[\sum_{j < a_i}f_{i,j} = \frac{(N-i)!}{\prod_{k = 1}^{3 \times 10^5}c_k}\sum_{k = 1}^{a_i-1} c_k
\]

然后没移动\(i\)一次，只会修改一个\(c_j\)，于是复杂度还是\(O(nlogn)\)。

但是还有一个问题——\(M\)不一定时素数。我们可以将其分解质因数

\[M = \prod_{i = 1}^qp_i^{d_i}
\]

然后我们只要能够计算出\(ans\)在模\(M_i = p_i^{d_i}\)的值，再通过中国剩余定理就可以计算答案了。那么这个怎么求呢？

我们可以将每个数字\(x\)用个二元组\((s,t)\)来表示，\(x = s \times p_i^t\)，且\((s,p_i) = 1\)。

于是有

• \((s,t) \times (u,v) = (s \times u,t+v)\)
• \((s,t) / (u,v) = (s \times u^{-1},t-v)\)

\(u^{-1}\)表示\(u\)在模\(M_i\)下的逆元。由此可以看出该二元组的第一关键字可以是模\(M_i\)意义下的。用此方法处理乘除。

当涉及到加法的时候将二元组转换成普通数即可。

#include<cstring>
#include<cstdio>
#include<cstdlib>
using namespace std;

typedef long long ll;
#define lowbit(a) (a&(-a))
#define maxn (300010)
int mod,ans,N,M,tot,A[maxn],aim[maxn],Mi[maxn],Pi[maxn],res[maxn],tree[maxn],num[maxn],tnum[maxn];

inline ll exgcd(ll a,ll b,ll c)
{
	if (!a) return -1;
	else if (!(c % a)) return c/a;
	ll t = exgcd(b % a,a,((-c % a)+a)%a);
	if (t == -1) return -1;
	return (t*b+c)/a;
}

inline ll qsm(ll a,int b,int c)
{
	ll ret = 1;
	for (;b;b >>= 1,(a *= a) %= c) if (b & 1) (ret *= a) %= c;
	return ret;
}

struct node
{
	int a,b;
	inline node(int x = 0,int p = 0) { if (!p) return; b = 0; while (!(x % p)) ++b,x /= p; a = x%mod; }
	friend inline node operator * (const node &x,const node &y)
	{
		node ret;
		ret.a = (ll)x.a*(ll)y.a%mod; ret.b = x.b+y.b;
		return ret;
	}
	friend inline node operator / (const node &x,const node &y)
	{
		node ret; int inv = exgcd(y.a,mod,1)%mod;
		ret.a = (ll)x.a*(ll)inv%mod; ret.b = x.b-y.b;
		return ret;
	}
	inline int tran(int p) { return (ll)a*qsm(p,b,mod)%mod; }
};

inline void ins(int a,int b) { for (;a <= 300000;a += lowbit(a)) tree[a] += b; }
inline int calc(int a) { int ret = 0; for (;a;a -= lowbit(a)) ret += tree[a]; return ret; }

inline void Div(int key)
{
	for (int i = 2;i*i <= key;++i)
		if (key % i == 0)
		{
			Mi[++tot] = 1; Pi[tot] = i;
			while (key % i == 0) Mi[tot] *= i,key /= i;
		}
	if (key > 1) Mi[++tot] = key,Pi[tot] = key;
}

inline void init()
{
	memset(tree,0,sizeof(tree)); memcpy(tnum,num,sizeof(num));
	for (int i = 1;i <= 300000;++i) if (num[i]) ins(i,num[i]);
}

inline void work(int id)
{
	init(); mod = Mi[id]; node now(1,Pi[id]);
	for (int i = 1;i < N;++i)
	{
		node tmp(i,Pi[id]);
		now = now*tmp;
	}
	for (int i = 1;i <= 300000;++i)
		for (int j = 2;j <= num[i];++j) { node tmp(j,Pi[id]); now = now/tmp; }
	for (int i = 1,sum;i <= N;++i)
	{
		if (sum = calc(A[i]-1)) res[id] += (now*node(sum,Pi[id])).tran(Pi[id]);
		if (res[id] >= mod) res[id] -= mod; ins(A[i],-1);
		if (i < N)
		{
			node tmp1(N-i,Pi[id]),tmp2(tnum[A[i]]--,Pi[id]);
			now = now*tmp2/tmp1;
		}
	}
}

inline int crt()
{
	int ret = 0;
	for (int i = 1;i <= tot;++i)
	{
		int tm = M/Mi[i],inv = exgcd(tm%Mi[i],Mi[i],1)%Mi[i];
		ret += ((ll)res[i]*(ll)inv%M*(ll)tm)%M;
		if (ret >= M) ret -= M;
	}
	return ret;
}

int main()
{
	// freopen("permutation.in","r",stdin);
	// freopen("permutation.out","w",stdout);
	scanf("%d %d",&N,&M);
	for (int i = 1;i <= N;++i) scanf("%d",A+i);
	Div(M);
	for (int i = 1;i <= N;++i) ++num[A[i]];
	for (int i = 1;i <= tot;++i) work(i);
	ans = crt(); if (++ans >= M) ans -= M;
	printf("%d",ans);
	// fclose(stdin); fclose(stdout);
	return 0;
}