[HAOI2018]染色

题面

神仙题。

令(lim=min{[n/s,m]}),容斥之后暴力推式子，有

[ans=m!n!sum_{i=0}^{lim}frac{(m-i)^{n-is}}{(s!)^i(m-i)!(n-is)!}sum_{i=0}^{j}frac{w_i}{i!}frac{(-1)^{j-i}}{(j-i)!} ]

(NTT)即可。

#include<bits/stdc++.h>
#include<algorithm>
#include<iostream>
#include<cstdlib>
#include<iomanip>
#include<cstring>
#include<complex>
#include<vector>
#include<cstdio>
#include<string>
#include<bitset>
#include<ctime>
#include<cmath>
#include<queue>
#include<stack>
#include<map>
#include<set>
#define FILE "a"
#define mp make_pair
#define pb push_back
#define RG register
#define il inline
using namespace std;
typedef unsigned long long ull;
typedef vector<int>VI;
typedef long long ll;
typedef double dd;
const dd eps=1e-10;
const int mod=1004535809;
const int N=10000010;
const dd pi=acos(-1);
const int inf=2147483647;
const ll INF=1e18+1;
il ll read(){
	RG ll data=0,w=1;RG char ch=getchar();
	while(ch!='-'&&(ch<'0'||ch>'9'))ch=getchar();
	if(ch=='-')w=-1,ch=getchar();
	while(ch<='9'&&ch>='0')data=data*10+ch-48,ch=getchar();
	return data*w;
}

il void file(){
	srand(time(NULL)+rand());
	freopen(FILE".in","r",stdin);
	freopen(FILE".out","w",stdout);
}

il int poww(int a,int b){
	RG int ret=1;
	for(;b;b>>=1,a=1ll*a*a%mod)
		if(b&1)ret=1ll*ret*a%mod;
	return ret;
}

int invf[N],fac[N];
il void init(){
	invf[0]=fac[0]=1;
	for(RG int i=1;i<N;i++)fac[i]=1ll*fac[i-1]*i%mod;
	invf[N-1]=poww(fac[N-1],mod-2);
	for(RG int i=N-2;i;i--)invf[i]=1ll*invf[i+1]*(i+1)%mod;
}

int l,r[N];
il void NTT(int *a,int n,int opt){
	for(l=0;(1<<l)<n;l++);n=(1<<l);
	for(RG int i=0;i<n;i++)r[i]=(r[i>>1]>>1)|((i&1)<<(l-1));
	for(RG int i=0;i<n;i++)if(i<r[i])swap(a[i],a[r[i]]);
	for(RG int i=2;i<=n;i<<=1){
		RG int wn=poww(opt==1?3:(mod+1)/3,(mod-1)/i);
		for(RG int j=0;j<n;j+=i){
			RG int w=1;
			for(RG int k=j;k<j+(i>>1);k++,w=1ll*w*wn%mod){
				RG int x=1ll*a[k+(i>>1)]*w%mod;
				a[k+(i>>1)]=(a[k]-x+mod)%mod;
				a[k]=(a[k]+x)%mod;
			}
		}
	}
	if(opt==-1)
		for(RG int i=0,rv=poww(n,mod-2);i<n;i++)
			a[i]=1ll*a[i]*rv%mod;
}

int n,m,s,lim,w[N],f[N],g[N],len,ans;
int main()
{
	n=read();m=read();s=read();lim=min(n/s,m);init();
	for(RG int i=0;i<=m;i++)w[i]=read();
	for(RG int i=0;i<=lim;i++)f[i]=1ll*w[i]*invf[i]%mod;
	for(RG int i=0;i<=lim;i++)g[i]=1ll*((i&1)?(mod-1):1)*invf[i]%mod;
	for(len=1;len<=(lim<<1);len<<=1);
	
	NTT(f,len,1);NTT(g,len,1);
	for(RG int i=0;i<len;i++)f[i]=1ll*f[i]*g[i]%mod;
	NTT(f,len,-1);
	
	for(RG int i=0;i<=lim;i++)
		(ans+=1ll*f[i]*poww(m-i,n-i*s)%mod*poww(invf[s],i)%mod*invf[m-i]%mod*invf[n-i*s]%mod)%=mod;
	printf("%lld
",1ll*ans*fac[n]%mod*fac[m]%mod);
	return 0;
}