C. New Year and Permutation

最新推荐文章于 2021-03-01 19:07:38 发布

原创最新推荐文章于 2021-03-01 19:07:38 发布 · 312 阅读

0 ·

CC 4.0 BY-SA版权

ACM 同时被 3 个专栏收录

359 篇文章

订阅专栏

codeforces

194 篇文章

订阅专栏

找规律

10 篇文章

订阅专栏

本文探讨了在给定长度和模数的情况下，计算所有可能排列的幸福总和的方法。幸福定义为排列中满足特定条件的子段数量，涉及到数学组合、算法优化和模运算。

链接：https://codeforces.ml/contest/1284/problem/C

Recall that the permutation is an array consisting of nn distinct integers from 11 to nn in arbitrary order. For example, [2,3,1,5,4][2,3,1,5,4] is a permutation, but [1,2,2][1,2,2] is not a permutation (22 appears twice in the array) and [1,3,4][1,3,4] is also not a permutation (n=3n=3 but there is 44 in the array).

A sequence aa is a subsegment of a sequence bb if aa can be obtained from bb by deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end. We will denote the subsegments as [l,r][l,r], where l,rl,r are two integers with 1≤l≤r≤n1≤l≤r≤n. This indicates the subsegment where l−1l−1 elements from the beginning and n−rn−r elements from the end are deleted from the sequence.

For a permutation p1,p2,…,pnp1,p2,…,pn, we define a framed segment as a subsegment [l,r][l,r] where max{pl,pl+1,…,pr}−min{pl,pl+1,…,pr}=r−lmax{pl,pl+1,…,pr}−min{pl,pl+1,…,pr}=r−l. For example, for the permutation (6,7,1,8,5,3,2,4)(6,7,1,8,5,3,2,4) some of its framed segments are: [1,2],[5,8],[6,7],[3,3],[8,8][1,2],[5,8],[6,7],[3,3],[8,8]. In particular, a subsegment [i,i][i,i] is always a framed segments for any ii between 11 and nn, inclusive.

We define the happiness of a permutation pp as the number of pairs (l,r)(l,r) such that 1≤l≤r≤n1≤l≤r≤n, and [l,r][l,r] is a framed segment. For example, the permutation [3,1,2][3,1,2] has happiness 55: all segments except [1,2][1,2] are framed segments.

Given integers nn and mm, Jongwon wants to compute the sum of happiness for all permutations of length nn, modulo the prime number mm. Note that there exist n!n! (factorial of nn) different permutations of length nn.

Input

The only line contains two integers nn and mm (1≤n≤2500001≤n≤250000, 108≤m≤109108≤m≤109, mm is prime).

Output

Print rr (0≤r<m0≤r<m), the sum of happiness for all permutations of length nn, modulo a prime number mm.

Examples

input

Copy

1 993244853

output

Copy

input

Copy

2 993244853

output

Copy

input

Copy

3 993244853

output

Copy

input

Copy

2019 993244853

output

Copy

923958830

input

Copy

2020 437122297

output

Copy

265955509

Note

For sample input n=3n=3, let's consider all permutations of length 33:

[1,2,3][1,2,3], all subsegments are framed segment. Happiness is 66.
[1,3,2][1,3,2], all subsegments except [1,2][1,2] are framed segment. Happiness is 55.
[2,1,3][2,1,3], all subsegments except [2,3][2,3] are framed segment. Happiness is 55.
[2,3,1][2,3,1], all subsegments except [2,3][2,3] are framed segment. Happiness is 55.
[3,1,2][3,1,2], all subsegments except [1,2][1,2] are framed segment. Happiness is 55.
[3,2,1][3,2,1], all subsegments are framed segment. Happiness is 66.

Thus, the sum of happiness is 6+5+5+5+5+6=326+5+5+5+5+6=32.

代码：

#include<bits/stdc++.h>
using namespace std;
long long n,mod,t,d,s,k;
long long l[1000001];
long long a[1000001];
long long f[1000001],e[1000001];
long long flag[1000001];
int main()
{
	cin>>n>>mod;
	a[0]=1;
	for(int i=1;i<=n;i++)
	{
		a[i]=a[i-1]*i%mod;
	}
	s=0;
	for(int i=1;i<=n;i++)
	{
		s+=(n-i+1)%mod*(a[i]*a[n-i+1]%mod);
		s%=mod;
	}
	cout<<s<<endl;
}