Submission #96536

#	Time	Username	Problem	Language	Result	Execution time	Memory
96536		qkxwsm	Energetic turtle (IZhO11_turtle)	C++14	90 / 100	2044 ms	85016 KiB

turtle

#pragma GCC optimize ("O3")
#pragma GCC target ("sse4")
#include <bits/stdc++.h>
#include <ext/pb_ds/tree_policy.hpp>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/rope>

using namespace std;
using namespace __gnu_pbds;
using namespace __gnu_cxx;

random_device(rd);
mt19937 rng(rd());
const long long FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count();

struct custom_hash
{
	template<class T>
	unsigned long long operator()(T v) const
	{
		unsigned long long x = v;
		x += FIXED_RANDOM;
		// x += 11400714819323198485ull;
		// x = (x ^ (x >> 30)) * 13787848793156543929ull;
		x = (x ^ (x >> 27)) * 10723151780598845931ull;
		return x ^ (x >> 31);
	}
};

template<class T> using ordered_set = tree<T, null_type, less<T>, rb_tree_tag, tree_order_statistics_node_update>;
template<class T, class U> using hash_table = gp_hash_table<T, U, custom_hash>;

template<class T>
T randomize(T mod)
{
	return (uniform_int_distribution<T>(0, mod - 1))(rng);
}
template<class T>
void readi(T &x)
{
	x = 0;
	bool negative = false;
	char c = ' ';
	while (c < '-')
	{
		c = getchar();
	}
	if (c == '-')
	{
		negative = true;
		c = getchar();
	}
	while (c >= '0')
	{
		x = x * 10 + (c - '0');
		c = getchar();
	}
	if (negative)
	{
		x = -x;
	}
}
template<class T>
void printi(T output)
{
	if (output == 0)
	{
		putchar('0');
		return;
	}
	if (output < 0)
	{
		putchar('-');
		output = -output;
	}
	int buf[20], n = 0;
	while(output)
	{
		buf[n] = ((output % 10));
		output /= 10;
		n++;
	}
	for (n--; n >= 0; n--)
	{
		putchar(buf[n] + '0');
	}
	return;
}
template<class T>
void ckmin(T &a, T b)
{
	a = min(a, b);
}
template<class T>
void ckmax(T &a, T b)
{
	a = max(a, b);
}
long long expo(long long a, long long e, long long mod)
{
	return ((e == 0) ? 1 : ((expo(a * a % mod, e >> 1, mod)) * ((e & 1) ? a : 1) % mod));
}
template<class T, class U>
void nmod(T &x, U mod)
{
	if (x >= mod) x -= mod;
}
template<class T>
T gcd(T a, T b)
{
	return (b ? gcd(b, a % b) : a);
}

#define y0 ___y0
#define y1 ___y1
#define MP make_pair
#define PB push_back
#define LB lower_bound
#define UB upper_bound
#define fi first
#define se second
#define DBG(x) cerr << #x << " = " << (x) << endl
#define SZ(x) ((int) ((x).size()))
#define FOR(i, a, b) for (auto i = (a); i < (b); i++)
#define FORD(i, a, b) for (auto i = (a) - 1; i >= (b); i--)
#define ALL(x) (x).begin(), (x).end()

const long double PI = 4.0 * atan(1.0);
const long double EPS = 1e-9;

#define MAGIC 347
#define SINF 10007
#define CO 1000007
#define INF 1000000007
#define BIG 1000000931
#define LARGE 1696969696967ll
#define GIANT 2564008813937411ll
#define LLINF 2696969696969696969ll
#define MAXN 600013

typedef long long ll;
typedef long double ld;
typedef pair<int, int> pii;
typedef pair<ll, ll> pll;
typedef pair<ld, ld> pdd;
typedef vector<int> vi;
typedef vector<ll> vl;
typedef vector<ld> vd;
typedef vector<pii> vpi;
typedef vector<pll> vpl;
typedef vector<pdd> vpd;

int N, M, K, T;
ll MOD;
pii coor[MAXN];
ll num[22];
ll dp[69][69];
ll ans = 0;
ll phi;
vpl primes;
hash_table<ll, vpl> factors;
pair<ll, vpl> fact[MAXN];
ll ifact[MAXN];
vl pows[MAXN];

ll choose(ll a, ll b)
{
	// cerr << a << ' ' << b << ' ' << dp[a][b] << endl;
	ll res = fact[a].fi;
	res *= ifact[b]; res %= MOD;
	res *= ifact[a - b]; res %= MOD;
	FOR(i, 0, SZ(primes))
	{
		// assert(fact[a].se[i].se - fact[a - b].se[i].se - fact[b].se[i].se >= 0);
		res *= pows[fact[a].se[i].se - fact[a - b].se[i].se - fact[b].se[i].se][i]; res %= MOD;
		// res *= expo(primes[i].fi, fact[a].se[i].se - fact[a - b].se[i].se - fact[b].se[i].se, MOD); res %= MOD;
	}
	// if (dp[a][b] != res)
	// {
	// 	cerr << "dead " << a << ' ' << b << ' ' << dp[a][b] << ' ' << res << endl; exit(0);
	// }
	return res;
	// cerr << a << ' ' << b << endl;
}

int32_t main()
{
	ios_base::sync_with_stdio(0); cin.tie(0);
	// cout << fixed << setprecision(10);
	// cerr << fixed << setprecision(10);
	// if (fopen("file.in", "r"))
	// {
	// 	freopen ("file.in", "r", stdin);
	// 	freopen ("file.out", "w", stdout);
	// }
	cin >> N >> M >> K >> T >> MOD;
	if (MOD == 1)
	{
		cout << "0\n";
		return 0;
	}
	int mod = MOD;
	for (int i = 2; i * i <= MOD; i++)
	{
		if (mod % i) continue;
		primes.PB({i, 0});
		while(mod % i == 0)
		{
			mod /= i;
			primes.back().se++;
		}
	}
	if (mod != 1)
	{
		primes.PB({mod, 1});
	}
	for (int i = 1; i * i <= MOD; i++)
	{
		if (MOD % i) continue;
		mod = i;
		for (pll p : primes)
		{
			factors[i].PB({p.fi, 0});
			if (i * i != MOD) factors[MOD / i].PB({p.fi, p.se});
			while(mod % p.fi == 0)
			{
				mod /= p.fi;
				factors[i].back().se++;
				if (i * i  != MOD) factors[MOD / i].back().se--;
			}
		}
	}
	// for (auto p : factors)
	// {
	// 	cerr << "factor " << p.fi << ":";
	// 	for (auto q : p.se)
	// 	{
	// 		cerr << ' ' << q.fi << ',' << q.se;
	// 	}
	// 	cerr << endl;
	// }
	FOR(i, 0, K)
	{
		cin >> coor[i].fi >> coor[i].se;
	}
	dp[0][0] = 1;
	FOR(i, 1, 42)
	{
		dp[i][0] = dp[i][i] = 1;
		FOR(j, 1, i) dp[i][j] = (dp[i - 1][j] + dp[i - 1][j - 1]) % MOD;
	}
	sort(coor, coor + K);
	FOR(i, 0, K + 1)
	{
		ll take = (i <= T ? 1 : 0) - num[i]; take += MOD; nmod(take, MOD);
		num[i] = take;
		FOR(j, i + 1, K + 1)
		{
			num[j] += take * dp[j][i];
			num[j] %= MOD; nmod(take, MOD);
		}
		// DBG(num[i]);
	}
	fact[0].fi = 1;
	fact[0].se = factors[1];
	FOR(i, 1, N + M + 2)
	{
		vpl f(SZ(primes));
		mod = i;
		FOR(j, 0, SZ(primes))
		{
			ll p = primes[j].fi;
			f[j].fi = p;
			while(mod % p == 0)
			{
				f[j].se++;
				mod /= p;
			}
		}
		fact[i].fi = (fact[i - 1].fi * mod) % MOD;
		fact[i].se = fact[i - 1].se;
		FOR(j, 0, SZ(fact[i].se))
		{
			fact[i].se[j].se += f[j].se;
		}
		// cerr << i << ' ' << fact[i].fi;
		// for (auto p : fact[i].se)
		// {
		// 	cerr << ' ' << p.fi << ',' << p.se;
		// }
		// cerr << endl;
	}
	phi = MOD;
	for (pll p : primes)
	{
		phi /= p.fi;
		phi *= (p.fi - 1);
	}
	FOR(i, 0, SZ(primes))
	{
		pows[0].PB(1);
	}
	FOR(i, 1, N + M + 2)
	{
		pows[i] = pows[i - 1];
		FOR(j, 0, SZ(primes))
		{
			pows[i][j] *= primes[j].fi;
			pows[i][j] %= MOD;
		}
	}
	FOR(i, 0, N + M + 2)
	{
		ifact[i] = expo(fact[i].fi, phi - 1, MOD);
	}
	// DBG(phi);
	FOR(mask, 0, (1 << K))
	{
		ll cur = num[__builtin_popcount(mask)];
		pii block = MP(0, 0);
		FOR(i, 0, K)
		{
			if (!(mask & (1 << i))) continue;
			if (coor[i].se < block.se)
			{
				cur = 0;
				break;
			}
			cur *= choose(coor[i].fi - block.fi + coor[i].se - block.se, coor[i].se - block.se);
			block = coor[i];
			cur %= MOD;
		}
		cur *= choose(N - block.fi + M - block.se, N - block.fi);
		cur %= MOD;
		// DBG(cur);
		ans += cur; nmod(ans, MOD);
	}
	cout << ans << '\n';
	// cerr << "time elapsed = " << (clock() / (CLOCKS_PER_SEC / 1000)) << " ms" << endl;
	return 0;
}
/* READ READ READ
* int overflow, maxn too small, special cases (n=1?, two distinct?), cin.tie() interactive
* reread the problem, try small cases
* note down possible sources of error as you go
* do smth instead of nothing
*/

• Subtask 1: 90 / 100