//~ mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
//~ while (clock()<=69*CLOCKS_PER_SEC)
//~ #pragma comment(linker, "/stack:200000000")
#pragma GCC optimize("O3")
//~ #pragma GCC target ("avx2")
//~ #pragma GCC optimize("Ofast")
//~ #pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native")
//~ #pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace __gnu_pbds;
using namespace std;
template <typename T>
using ordered_set =
tree<T, null_type, less<T>, rb_tree_tag, tree_order_statistics_node_update>;
#define sim template < class c
#define ris return * this
#define dor > debug & operator <<
#define eni(x) sim > typename \
enable_if<sizeof dud<c>(0) x 1, debug&>::type operator<<(c i) {
sim > struct rge { c b, e; };
sim > rge<c> range(c i, c j) { return rge<c>{i, j}; }
sim > auto dud(c* x) -> decltype(cerr << *x, 0);
sim > char dud(...);
struct debug {
#ifdef LOCAL
~debug() { cerr << endl; }
eni(!=) cerr << boolalpha << i; ris; }
eni(==) ris << range(begin(i), end(i)); }
sim, class b dor(pair < b, c > d) {
ris << "(" << d.first << ", " << d.second << ")";
}
sim dor(rge<c> d) {
*this << "[";
for (auto it = d.b; it != d.e; ++it)
*this << ", " + 2 * (it == d.b) << *it;
ris << "]";
}
#else
sim dor(const c&) { ris; }
#endif
};
#define imie(...) " [" << #__VA_ARGS__ ": " << (__VA_ARGS__) << "] "
#define shandom_ruffle random_shuffle
using ll=long long;
#define int ll
using pii=pair<int,int>;
using pll=pair<ll,ll>;
using vi=vector<int>;
using vll=vector<ll>;
const int nax=5000*1007;
ll mod;
ll inv(ll v)
{
if (v<=1)
return v;
return inv(mod%v)*(mod-mod/v)%mod;
}
ll sil[nax];
ll binom(int a, int b)
{
if (b>a || b<0)
return 0;
return sil[a]*inv(sil[b]*sil[a-b]%mod)%mod;
}
ll regular_shape(vi wek)
{
int ile=0;
ll mi=1;
for (int i=0; i<(int)wek.size(); i++)
{
ile+=wek[i];
int wsk=(int)wek.size()-1;
for (int j=1; j<=wek[i]; j++)
{
while(wek[wsk]<j)
wsk--;
mi=mi*(wek[i]-j+wsk-i+1)%mod;
}
}
return sil[ile]*inv(mi)%mod;
}
ll pot(ll a, ll b)
{
ll ret=1;
while(b)
{
if (b&1)
ret=(ret*a)%mod;
a=(a*a)%mod;
b>>=1;
}
return ret;
}
ll mysterious_e(ll r, ll p, ll s)
{
if (!(r&1))
{
ll ret=1;
for (ll l=r+1; l<2*p-r+2; l++)
ret=ret*(l+2*s)%mod;
ret=pot(ret, r/2);
for (ll l=2; l<=r; l++)
{
ll x=(l+2*s)*(2*p-l+2*s+2)%mod;
ret=ret*pot(x, l/2)%mod;
}
return inv(ret);
}
else
{
return sil[(r-1)/2+s]*inv(sil[p-(r-1)/2+s])%mod*mysterious_e(r-1, p, s)%mod;
}
}
ll shifted_shape(vi wek)
{
int ile=0;
for (int i : wek)
ile+=i;
ll li=sil[ile];
ll mi=1;
for (int i : wek)
mi=mi*sil[i]%mod;
for (int i=0; i<(int)wek.size(); i++)
{
for (int j=i+1; j<(int)wek.size(); j++)
{
li=li*(wek[i]-wek[j])%mod;
mi=mi*(wek[i]+wek[j])%mod;
}
}
return li*inv(mi)%mod;
}
ll rectangle_truncated_by_staircase_shape(int m, int n, int k)
{
ll N=m*n-k*(k+1)/2;
ll ret=1;
ret=ret*binom(N, m*(n-k-1))%mod;
ret=ret*regular_shape(vi(n-k-1, m))%mod;
vi wek;
for (int i=m; i>=m-k; i--)
wek.push_back(i);
ret=ret*shifted_shape(wek)%mod;
ret=ret*mysterious_e(k+1, m, n-k-1)%mod;
ret=ret*inv(mysterious_e(k+1, m, 0))%mod;
return ret;
}
signed main()
{
int n;
int A, B, C;
int brak;
scanf("%lld%lld%lld", &A, &B, &C);
scanf("%lld", &mod);
sil[0]=1;
for (int i=1; i<nax; i++)
sil[i]=sil[i-1]*i%mod;
n=A*B;
brak=A+B-C-1;
for (int i=1; i<=brak; i++)
n-=i;
vi wek(B, A);
for (int i=1; i<=brak; i++)
wek[B-i]-=(brak+1-i);
ll w1=regular_shape(wek);
ll w2=rectangle_truncated_by_staircase_shape(B, A, brak);
ll wyn=(w1*w2)%mod;
printf("%lld %lld\n", n, wyn);
return 0;
}
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 | //~ mt19937 rng(chrono::steady_clock::now().time_since_epoch().count()); //~ while (clock()<=69*CLOCKS_PER_SEC) //~ #pragma comment(linker, "/stack:200000000") #pragma GCC optimize("O3") //~ #pragma GCC target ("avx2") //~ #pragma GCC optimize("Ofast") //~ #pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native") //~ #pragma GCC optimize("unroll-loops") #include <bits/stdc++.h> #include <ext/pb_ds/assoc_container.hpp> #include <ext/pb_ds/tree_policy.hpp> using namespace __gnu_pbds; using namespace std; template <typename T> using ordered_set = tree<T, null_type, less<T>, rb_tree_tag, tree_order_statistics_node_update>; #define sim template < class c #define ris return * this #define dor > debug & operator << #define eni(x) sim > typename \ enable_if<sizeof dud<c>(0) x 1, debug&>::type operator<<(c i) { sim > struct rge { c b, e; }; sim > rge<c> range(c i, c j) { return rge<c>{i, j}; } sim > auto dud(c* x) -> decltype(cerr << *x, 0); sim > char dud(...); struct debug { #ifdef LOCAL ~debug() { cerr << endl; } eni(!=) cerr << boolalpha << i; ris; } eni(==) ris << range(begin(i), end(i)); } sim, class b dor(pair < b, c > d) { ris << "(" << d.first << ", " << d.second << ")"; } sim dor(rge<c> d) { *this << "["; for (auto it = d.b; it != d.e; ++it) *this << ", " + 2 * (it == d.b) << *it; ris << "]"; } #else sim dor(const c&) { ris; } #endif }; #define imie(...) " [" << #__VA_ARGS__ ": " << (__VA_ARGS__) << "] " #define shandom_ruffle random_shuffle using ll=long long; #define int ll using pii=pair<int,int>; using pll=pair<ll,ll>; using vi=vector<int>; using vll=vector<ll>; const int nax=5000*1007; ll mod; ll inv(ll v) { if (v<=1) return v; return inv(mod%v)*(mod-mod/v)%mod; } ll sil[nax]; ll binom(int a, int b) { if (b>a || b<0) return 0; return sil[a]*inv(sil[b]*sil[a-b]%mod)%mod; } ll regular_shape(vi wek) { int ile=0; ll mi=1; for (int i=0; i<(int)wek.size(); i++) { ile+=wek[i]; int wsk=(int)wek.size()-1; for (int j=1; j<=wek[i]; j++) { while(wek[wsk]<j) wsk--; mi=mi*(wek[i]-j+wsk-i+1)%mod; } } return sil[ile]*inv(mi)%mod; } ll pot(ll a, ll b) { ll ret=1; while(b) { if (b&1) ret=(ret*a)%mod; a=(a*a)%mod; b>>=1; } return ret; } ll mysterious_e(ll r, ll p, ll s) { if (!(r&1)) { ll ret=1; for (ll l=r+1; l<2*p-r+2; l++) ret=ret*(l+2*s)%mod; ret=pot(ret, r/2); for (ll l=2; l<=r; l++) { ll x=(l+2*s)*(2*p-l+2*s+2)%mod; ret=ret*pot(x, l/2)%mod; } return inv(ret); } else { return sil[(r-1)/2+s]*inv(sil[p-(r-1)/2+s])%mod*mysterious_e(r-1, p, s)%mod; } } ll shifted_shape(vi wek) { int ile=0; for (int i : wek) ile+=i; ll li=sil[ile]; ll mi=1; for (int i : wek) mi=mi*sil[i]%mod; for (int i=0; i<(int)wek.size(); i++) { for (int j=i+1; j<(int)wek.size(); j++) { li=li*(wek[i]-wek[j])%mod; mi=mi*(wek[i]+wek[j])%mod; } } return li*inv(mi)%mod; } ll rectangle_truncated_by_staircase_shape(int m, int n, int k) { ll N=m*n-k*(k+1)/2; ll ret=1; ret=ret*binom(N, m*(n-k-1))%mod; ret=ret*regular_shape(vi(n-k-1, m))%mod; vi wek; for (int i=m; i>=m-k; i--) wek.push_back(i); ret=ret*shifted_shape(wek)%mod; ret=ret*mysterious_e(k+1, m, n-k-1)%mod; ret=ret*inv(mysterious_e(k+1, m, 0))%mod; return ret; } signed main() { int n; int A, B, C; int brak; scanf("%lld%lld%lld", &A, &B, &C); scanf("%lld", &mod); sil[0]=1; for (int i=1; i<nax; i++) sil[i]=sil[i-1]*i%mod; n=A*B; brak=A+B-C-1; for (int i=1; i<=brak; i++) n-=i; vi wek(B, A); for (int i=1; i<=brak; i++) wek[B-i]-=(brak+1-i); ll w1=regular_shape(wek); ll w2=rectangle_truncated_by_staircase_shape(B, A, brak); ll wyn=(w1*w2)%mod; printf("%lld %lld\n", n, wyn); return 0; } |
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