[BZOJ5306][HAOI2018]染色(容斥+FFT)

https://www.cnblogs.com/zhoushuyu/p/9138251.html

注意如果一开始F(i)中内层式子中j枚举的是除前i种颜色之外还有几种出现S次的颜色，那么后面式子就会难推很多。

#include<cstdio>
#include<algorithm>
#define rep(i,l,r) for (int i=(l); i<=(r); i++)
using namespace std;

const int N=300010,M=10000010,mod=1004535809;
int n,m,s,ans,w[N],fac[M],inv[M],rev[N],a[N],b[N];

int ksm(int a,int b){
    int res=1;
    for (; b; a=1ll*a*a%mod,b>>=1)
        if (b & 1) res=1ll*res*a%mod;
    return res;
}

void NTT(int a[],int n,bool f){
    for (int i=0; i<n; i++) if (i<rev[i]) swap(a[i],a[rev[i]]);
    for (int i=1; i<n; i<<=1){
        int wn=ksm(3,f ? (mod-1)/(i<<1) : (mod-1)-(mod-1)/(i<<1));
        for (int p=i<<1,j=0; j<n; j+=p){
            int w=1;
            for (int k=0; k<i; k++,w=1ll*w*wn%mod){
                int x=a[j+k],y=1ll*w*a[i+j+k]%mod;
                a[j+k]=(x+y)%mod; a[i+j+k]=(x-y+mod)%mod;
            }
        }
    }
    if (f) return;
    int inv=ksm(n,mod-2);
    for (int i=0; i<n; i++) a[i]=1ll*a[i]*inv%mod;
}

int main(){
    freopen("color.in","r",stdin);
    freopen("color.out","w",stdout);
    scanf("%d%d%d",&n,&m,&s); int N=min(m,n/s),ed=max(n,m);
    rep(i,0,m) scanf("%d",&w[i]);
    fac[0]=1; rep(i,1,ed) fac[i]=1ll*fac[i-1]*i%mod;
    inv[ed]=ksm(fac[ed],mod-2);
    for (int i=ed-1; ~i; i--) inv[i]=1ll*inv[i+1]*(i+1)%mod;
    rep(i,0,N) a[i]=1ll*w[i]*inv[i]%mod;
    rep(i,0,N) b[i]=(i&1)?mod-inv[i]:inv[i];
    int nn=1,L=0; for (; nn<=2*N; nn<<=1,L++);
    for (int i=0; i<nn; i++) rev[i]=(rev[i>>1]>>1)|((i&1)<<(L-1));
    NTT(a,nn,1); NTT(b,nn,1);
    for (int i=0; i<nn; i++) a[i]=1ll*a[i]*b[i]%mod;
    NTT(a,nn,0);
    rep(i,0,N) ans=(ans+1ll*ksm(m-i,n-i*s)*inv[m-i]%mod*ksm(inv[s],i)%mod*inv[n-i*s]%mod*a[i]%mod)%mod;
    printf("%lld
",1ll*ans*fac[n]%mod*fac[m]%mod);
    return 0;
}