题解 queen(留坑)
博客园突然打不开了,奇奇怪怪的……
少写个等号没看出来 nm写反了没看出来 考完5min全拍出来了
手残属性加持 不对拍等于爆零
yysy,我连卢卡斯定理的存在都忘了……
发现要让一大堆皇后能互相攻击,它们貌似只能在同一条直线上
然后发现皇后数量较少的时候好像有特例
所以特判即可
\(O(n)\)解法需要枚举边长,考虑如何优化
枚举边长是省不掉的,考虑处理下柿子
\[\sum\limits_{i=1}^{min(n,m)-1}(n-i)(m-i) = \sum\limits_{i=1}^{min(n,m)-1}nm-(n+m)+i^2 = (min(n,m)-1)nm-(n+m)\sum\limits_{i=1}^{min(n,m)-1}i+\sum\limits_{i=1}^{min(n,m)-1}i^2
\]
- 有个式子:\(\sum\limits_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}\)
同理,[留坑]
然后还有一个需要处理的式子是 \(\sum\limits_{i=k}^n C_i^k\)
\[\sum\limits_{i=k}^n C_i^k = C_{k+1}^{k+1}+\sum\limits_{i=k+1}^n C_i^k = C_{k+1}^{k+1}+C_{k+1}^k+\sum\limits_{i=k+2}^n C_i^k
\]
发现这两项可以组合,即为
\[C_{k+1}^k + \sum\limits_{i=k+2}^n C_i^k
\]
依次组合,最终原式等于 \(C_{n+1}^{k+1}\)
- 结论:\(\sum\limits_{i=0}^n C_i^k = C_{n+1}^{k+1}\)
Code:
#include <bits/stdc++.h>
using namespace std;
#define INF 0x3f3f3f3f
#define N 310010
#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 ll read() {
ll 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;
}
ll n, m, k, kn, km;
ll fac[N], inv1[N], inv2[N];
const ll p=3e5+7;
inline ll C(ll n, ll k) {return n>=k?fac[n]*inv2[k]%p*inv2[n-k]%p:0;}
inline ll lucas(ll n, ll k) {return !k?1:lucas(n/p, k/p)*C(n%p, k%p)%p;}
namespace task1{
inline ll sig(ll n) {n%=p; return n*(n+1)*inv1[2]%p;}
inline ll sig2(ll n) {n%=p; return n*(n+1)%p*(n*2+1)%p*inv1[6]%p;}
void solve() {
ll ans=0, lim;
n=read(); m=read(); k=read();
kn=n%p; km=m%p;
if (k==1) {printf("%lld\n", (kn*km)%p); return ;}
if (k==3) {
lim=(min(n, m)-1)%p; ans+=4ll*(lim*kn%p*km%p - (kn+km)%p*sig(lim)%p + sig2(lim))%p, ans%=p;
lim=min((n-1)/2, m-1)%p; ans+=2ll*(lim*kn%p*km%p - (2ll*km+kn)%p*sig(lim)%p + 2ll*sig2(lim))%p, ans%=p;
lim=min(n-1, (m-1)/2)%p; ans+=2ll*(lim*kn%p*km%p - (2ll*kn+km)%p*sig(lim)%p + 2ll*sig2(lim))%p, ans%=p;
}
if (k==4) {
lim=min((n-1)/2, m-1)%p; ans+=2ll*(lim*kn%p*km%p - (2ll*km+kn)%p*sig(lim)%p + 2ll*sig2(lim))%p, ans%=p;
lim=min(n-1, (m-1)/2)%p; ans+=2ll*(lim*kn%p*km%p - (2ll*kn+km)%p*sig(lim)%p + 2ll*sig2(lim))%p, ans%=p;
lim=(min(n, m)-1)%p; ans+=(lim*kn%p*km%p - (kn+km)%p*sig(lim)%p + sig2(lim))%p, ans%=p;
lim=((min(n, m)-1)/2)%p; ans+=5ll*(lim*kn%p*km%p - 2ll*(kn+km)%p*sig(lim)%p + 4ll*sig2(lim))%p, ans%=p;
}
if (k==5) {lim=((min(n, m)-1)/2)%p; ans+=2ll*(lim*kn%p*km%p - 2ll*(kn+km)%p*sig(lim)%p + 4ll*sig2(lim))%p, ans%=p;}
if (m>=k) ans+=kn*lucas(m, k)%p, ans%=p;
if (n>=k) ans+=km*lucas(n, k)%p, ans%=p;
if (n>=k && m>=k) {
ans+=4ll*lucas(min(n, m), k+1)%p, ans%=p;
ans+=2ll*(max(n, m)-min(n, m)+1)%p*lucas(min(n, m), k)%p, ans%=p;
}
printf("%lld\n", (ans%p+p)%p);
}
}
signed main()
{
int T;
T=read();
fac[0]=fac[1]=1; inv1[0]=inv1[1]=1; inv2[0]=inv2[1]=1;
for (int i=2; i<N; ++i) fac[i]=fac[i-1]*i%p;
for (int i=2; i<N; ++i) inv1[i]=(p-p/i)*inv1[p%i]%p;
for (int i=2; i<N; ++i) inv2[i]=inv1[i]*inv2[i-1]%p;
while (T--) task1::solve();
return 0;
}