题解 解方程
状压枚举有哪些变量不满足要求，借此完成容斥
然后就是exLucas板子了
但我显然并不记得这个板子了
点击查看代码
#include <bits/stdc++.h>
using namespace std;
#define INF 0x3f3f3f3f
#define N 100010
#define ll long long
//#define int long long
char buf[1<<21], *p1=buf, *p2=buf;
#define getchar() (p1==p2&&(p2=(p1=buf)+fread(buf, 1, 1<<21, stdin)), p1==p2?EOF:*p1++)
inline int read() {
int ans=0, f=1; char c=getchar();
while (!isdigit(c)) {if (c=='-') f=-f; c=getchar();}
while (isdigit(c)) {ans=(ans<<3)+(ans<<1)+(c^48); c=getchar();}
return ans*f;
}
int n, n1, n2, m;
ll a[N], mod;
namespace task1{
bool inited;
ll fac[N], inv[N], f[N];
inline ll C(int n, int k) {return n<k?0:fac[n]*inv[k]%mod*inv[n-k]%mod;}
inline ll lucas(ll n, ll k) {return !n?1:lucas(n/mod, k/mod)*C(n%mod, k%mod)%mod;}
void init() {
fac[0]=fac[1]=1; inv[0]=inv[1]=1;
for (int i=2; i<mod; ++i) fac[i]=fac[i-1]*i%mod;
for (int i=2; i<mod; ++i) inv[i]=(mod-mod/i)*inv[mod%i]%mod;
for (int i=2; i<mod; ++i) inv[i]=inv[i-1]*inv[i]%mod;
}
ll calc(int s) {
ll ans=0, dlt=0;
for (int i=1; i<=n1; ++i) if (s&(1<<(i-1))) dlt+=a[i];
if (m-dlt-1>=n-1) ans=(ans+lucas(m-dlt-1, n-1))%mod;
// cout<<"dlt: "<<dlt<<' '<<m-dlt-1<<' '<<n-1<<endl;
return ans;
}
void solve() {
if (!inited) init(), inited=1;
for (int i=n1+1; i<=n1+n2; ++i) m-=a[i]-1;
// cout<<"m: "<<m<<endl;
if (m<n) {puts("0"); return ;}
int lim=1<<n1;
for (int i=0; i<=n1; ++i) f[i]=0;
for (int s=0,cnt; s<lim; ++s) {
cnt=__builtin_popcount(s);
f[cnt]=(f[cnt]+calc(s))%mod;
}
// cout<<"f: "; for (int i=0; i<=n1; ++i) cout<<f[i]<<' '; cout<<endl;
ll ans=0;
for (int i=0; i<=n1; ++i) ans=(ans+(i&1?-1:1)*f[i])%mod;
printf("%lld\n", (ans%mod+mod)%mod);
}
}
namespace task2{
bool npri[N], inited;
ll f[N], r[N], tm[N], tem[N];
int pri[N], low[N], lowp[N], lowc[N], pcnt, div[N], dcnt;
inline ll qpow(ll a, ll b, ll mod) {ll ans=1; for (; b; a=a*a%mod,b>>=1) if (b&1) ans=ans*a%mod; return ans;}
void exgcd(ll a, ll b, ll& x, ll& y) {
if (!b) {x=1; y=0; return ;}
exgcd(b, a%b, y, x);
y-=a/b*x;
}
void init() {
ll t=mod; dcnt=0;
for (int i=2; i*i<=mod; ++i) if (t%i==0) {
div[++dcnt]=1;
do {t/=i; div[dcnt]*=i;} while (t%i==0);
}
if (t>1) div[++dcnt]=t;
// cout<<"div: "; for (int i=1; i<=dcnt; ++i) cout<<div[i]<<' '; cout<<endl;
for (int i=2; i<N; ++i) {
if (!npri[i]) pri[++pcnt]=low[i]=lowp[i]=i, lowc[i]=1;
for (int j=1,x; j<=pcnt&&i*pri[j]<N; ++j) {
npri[x=i*pri[j]]=1;
if (!(i%pri[j])) {
low[x]=pri[j];
lowp[x]=lowp[i]*pri[j];
lowc[x]=lowc[i]+1;
break;
}
else low[x]=lowp[x]=pri[j], lowc[x]=1;
}
}
}
void divide(int i) {
for (int t=i; t>1; t/=lowp[t]) tem[low[t]]+=tem[i]*lowc[t];
tem[i]=0;
}
ll crt(int n) {
ll M=1, ans=0;
for (int i=1; i<=n; ++i) M*=tm[i];
for (int i=1; i<=n; ++i) {
// cout<<"i: "<<i<<endl;
ll w=M/tm[i], x, y;
exgcd(w, tm[i], x, y);
// cout<<"x: "<<x<<' '<<w*x%m[i]<<endl;
ans=(ans+x*r[i]*w)%M;
}
return (ans%M+M)%M;
}
void multi_fac(ll n, ll op, ll p) {
// cout<<"multi: "<<n<<' '<<op<<' '<<p<<endl;
ll t=n;
while (t) {
// cout<<"t: "<<t<<endl;
ll cnt=t/p;
for (int i=1; i<=p; ++i) tem[i]+=op*cnt;
for (int i=1; i<=t%p; ++i) tem[i]+=op;
t/=p;
}
}
ll exlucas(ll n, ll k) {
for (int i=1; i<=dcnt; ++i) {
for (int j=0; j<=div[i]; ++j) tem[j]=0;
multi_fac(n, 1, div[i]);
multi_fac(k, -1, div[i]);
multi_fac(n-k, -1, div[i]);
for (int j=div[i]; j>1; --j) if (npri[j]) divide(j); //, cout<<"divide: "<<j<<endl;
// cout<<"tem: "; for (int j=1; j<=div[i]; ++j) cout<<tem[j]<<' '; cout<<endl;
r[i]=1; tm[i]=div[i];
for (int j=2; j<=div[i]; ++j)
if (tem[j]>0) r[i]=r[i]*qpow(j, tem[j], div[i])%div[i];
else if (tem[j]<0) r[i]=r[i]*qpow(qpow(j, div[i]-2, div[i]), -tem[j], div[i])%div[i];
}
// cout<<"crt_r: "; for (int i=1; i<=dcnt; ++i) cout<<r[i]<<' '; cout<<endl;
// cout<<"crt_m: "; for (int i=1; i<=dcnt; ++i) cout<<tm[i]<<' '; cout<<endl;
return crt(dcnt);
}
ll calc(int s) {
ll ans=0, dlt=0;
for (int i=1; i<=n1; ++i) if (s&(1<<(i-1))) dlt+=a[i];
if (m-dlt-1>=n-1) ans=(ans+exlucas(m-dlt-1, n-1))%mod;
// cout<<"dlt: "<<dlt<<' '<<m-dlt-1<<' '<<n-1<<endl;
return ans;
}
void solve() {
if (!inited) init(), inited=1;
for (int i=n1+1; i<=n1+n2; ++i) m-=a[i]-1;
// cout<<"m: "<<m<<endl;
if (m<n) {puts("0"); return ;}
int lim=1<<n1;
for (int i=0; i<=n1; ++i) f[i]=0;
for (int s=0,cnt; s<lim; ++s) {
cnt=__builtin_popcount(s);
f[cnt]=(f[cnt]+calc(s))%mod;
}
// cout<<"f: "; for (int i=0; i<=n1; ++i) cout<<f[i]<<' '; cout<<endl;
ll ans=0;
for (int i=0; i<=n1; ++i) ans=(ans+(i&1?-1:1)*f[i])%mod;
printf("%lld\n", (ans%mod+mod)%mod);
}
}
namespace task{
bool npri[N], inited;
ll f[N], r[N], tm[N], tem[N];
int pri[N], low[N], lowp[N], lowc[N], pcnt, div[N], pr[N], dcnt;
inline ll qpow(ll a, ll b, ll mod) {ll ans=1; for (; b; a=a*a%mod,b>>=1) if (b&1) ans=ans*a%mod; return ans;}
void exgcd(ll a, ll b, ll& x, ll& y) {
if (!b) {x=1; y=0; return ;}
exgcd(b, a%b, y, x);
y-=a/b*x;
}
void init() {
ll t=mod; dcnt=0;
for (int i=2; i*i<=mod; ++i) if (t%i==0) {
div[++dcnt]=1; pr[dcnt]=i;
do {t/=i; div[dcnt]*=i;} while (t%i==0);
}
if (t>1) div[++dcnt]=t, pr[dcnt]=t;
// cout<<"div: "; for (int i=1; i<=dcnt; ++i) cout<<div[i]<<' '; cout<<endl;
for (int i=2; i<N; ++i) {
if (!npri[i]) pri[++pcnt]=low[i]=lowp[i]=i, lowc[i]=1;
for (int j=1,x; j<=pcnt&&i*pri[j]<N; ++j) {
npri[x=i*pri[j]]=1;
if (!(i%pri[j])) {
low[x]=pri[j];
lowp[x]=lowp[i]*pri[j];
lowc[x]=lowc[i]+1;
break;
}
else low[x]=lowp[x]=pri[j], lowc[x]=1;
}
}
}
void divide(int i) {
for (int t=i; t>1; t/=lowp[t]) tem[low[t]]+=tem[i]*lowc[t];
tem[i]=0;
}
ll inv(ll n, ll p) {
if (!n) return 0;
ll x, y;
exgcd(n, p, x, y);
x=(x%p+p)%p;
if (!x) x+=p;
return x;
}
ll crt(int n) {
ll M=1, ans=0;
for (int i=1; i<=n; ++i) M*=tm[i];
for (int i=1; i<=n; ++i) {
// cout<<"i: "<<i<<endl;
ll w=M/tm[i], x, y;
exgcd(w, tm[i], x, y);
// cout<<"x: "<<x<<' '<<w*x%m[i]<<endl;
ans=(ans+x*r[i]*w)%M;
}
return (ans%M+M)%M;
}
ll fac(ll n, ll x, ll p) {
if (!n) return 1;
ll s=1;
for (int i=1; i<=p; ++i) if (i%x) s=s*i%p;
s=qpow(s, n/p, p);
for (int i=n/p*p+1; i<=n; ++i) if (i%x) s=s*i%p;
return s*fac(n/x, x, p)%p;
}
ll multilucas(ll n, ll k, ll x, ll p) {
ll cnt=0;
for (ll i=n; i; i/=x) cnt+=i/x;
for (ll i=k; i; i/=x) cnt-=i/x;
for (ll i=n-k; i; i/=x) cnt-=i/x;
return qpow(x, cnt, p)*fac(n, x, p)%p*inv(fac(k, x, p), p)%p*inv(fac(n-k, x, p), p)%p;
}
ll exlucas(ll n, ll k) {
for (int i=1; i<=dcnt; ++i) r[i]=multilucas(n, k, pr[i], div[i]), tm[i]=div[i];
// cout<<"crt_r: "; for (int i=1; i<=dcnt; ++i) cout<<r[i]<<' '; cout<<endl;
// cout<<"crt_m: "; for (int i=1; i<=dcnt; ++i) cout<<tm[i]<<' '; cout<<endl;
return crt(dcnt);
}
ll calc(int s) {
ll ans=0, dlt=0;
for (int i=1; i<=n1; ++i) if (s&(1<<(i-1))) dlt+=a[i];
if (m-dlt-1>=n-1) ans=(ans+exlucas(m-dlt-1, n-1))%mod;
// cout<<"dlt: "<<dlt<<' '<<m-dlt-1<<' '<<n-1<<endl;
return ans;
}
void solve() {
if (!inited) init(), inited=1;
for (int i=n1+1; i<=n1+n2; ++i) m-=a[i]-1;
// cout<<"m: "<<m<<endl;
if (m<n) {puts("0"); return ;}
int lim=1<<n1;
for (int i=0; i<=n1; ++i) f[i]=0;
for (int s=0,cnt; s<lim; ++s) {
cnt=__builtin_popcount(s);
f[cnt]=(f[cnt]+calc(s))%mod;
}
// cout<<"f: "; for (int i=0; i<=n1; ++i) cout<<f[i]<<' '; cout<<endl;
ll ans=0;
for (int i=0; i<=n1; ++i) ans=(ans+(i&1?-1:1)*f[i])%mod;
printf("%lld\n", (ans%mod+mod)%mod);
}
}
signed main()
{
int T=read(); mod=read();
while (T--) {
n=read(); n1=read(); n2=read(); m=read();
for (int i=1; i<=n1+n2; ++i) a[i]=read();
// if (mod==10007) task1::solve();
// else task2::solve();
task::solve();
}
// mod=7;
// task2::init();
// cout<<task2::exlucas(4, 2)<<endl;
// int n=read(), k=read(); mod=read();
// task2::init();
// cout<<task2::exlucas(n, k)<<endl;
return 0;
}
