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Commit 89858b9

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Add sieve and combinatorics to math
1 parent b8df5af commit 89858b9

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4 files changed

+147
-10
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4 files changed

+147
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‎Library/Math/Combinatorics.cpp

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#include <bits/stdc++.h>
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#include "fexp.cpp" //LATEX_IGNORED_LINE
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#define ll long long
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using namespace std;
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struct Combin {
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vector<ll> fat, finv;
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Combin(int n){
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fat.assign(n+1, 1);
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for(int i=2; i<=n; i++) fat[i] = fat[i-1]*i % mod;
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finv.assign(n+1, fexp(fat.back(), mod-2));
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for(int i=n; i>0; i--) finv[i-1] = finv[i]*i % mod;
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}
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ll choose(ll n, ll k){ assert(n < fat.size()); return k>n||k<0 ? 0 : fat[n] * finv[k] % mod * finv[n-k] % mod; } //precalc O(N)
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ll chooseLinear(ll n, ll k){ //O(k) || min(k, n-k);
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k = min(k, n-k);
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ll ans = 1, inv=1;
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for(int i=n; i>k; i--) ans = ans*i % mod;
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for(int i=1; i<=n-k; i++) inv = inv*i % mod;
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return ans * fexp(inv, mod-2) % mod;
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}
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ll permRepetition(const vector<int> &cnt){
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ll n = accumulate(begin(cnt), end(cnt), 0ll), ans = fat[n];
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for(int x : cnt) ans = ans * finv[x] % mod;
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return ans;
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}
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ll sumNci (ll n){ return fexp(2, n); } //for(i=0; i<=n) sum+=choose(n, i);
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ll sumicK (ll n, ll k){ return choose(n+1, k+1); } //for(i=0; i<=n) sum+=choose(i, k);
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ll sumNKcK(ll n, ll k){ return choose(n+k+1, k); } //for(i=0; i<=k) sum+=choose(n+i, i);
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ll sumNsqr(ll n){ return choose(n+n, n); } //for(i=0; i<=n) sum += pow(choose(n, i), 2);
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ll catalan(ll n){ return choose(2*n, n) * fexp(n+1, mod-2) % mod; }
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// Stars and Bars
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ll starsBars(ll n, ll k){ return choose(n+k-1, n); } //O(choose)
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ll starsLowerBound(ll n, const vector<ll> &lw){ //O(k)
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for(auto x : lw) n -= x;
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return starsBars(n, lw.size());
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}
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ll starsUpperBound(ll n, ll k, ll up){ //O(k)
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ll ans = 0;
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for(int i=0; i<=k; i++)
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ans += choose(k, i) * choose(n+k-1-(up+1)*i, k-1) % mod * (i&1? -1:+1);
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return ans;
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}
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ll starsUpperBound(ll M, const vector<ll> &up){ //O(N*M)
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int N = up.size();
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vector dp(up.size()+1, vector<ll>(N+1));
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for(int m=0; m<=M; m++) dp[0][m] = choose(N+m-1, m);
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for(int n=1; n<=N; n++)
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for(int m=0; m<=M; m++)
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dp[n][m] = dp[n-1][m] - (m-up[n-1]-1 < 0 ? 0 : dp[n-1][m-up[n-1]-1]);
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return dp[N][M];
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}
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ll starsLowerUpperBound(ll n, const vector<ll> &lw, const vector<ll> &up){ //O(N*M)
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for(auto x : lw) n -= x;
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return starsUpperBound(n, up);
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}
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};
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const int MAXN = 5e3;
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ll pascal[MAXN][MAXN];
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// pascal[n][k] = choose(n, k);
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void Pascal(int N){
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pascal[0][0] = 1;
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for(int n=1; n<=N; n++){
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pascal[n][0] = pascal[n][n] = 1;
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for(int k=1; k<n; k++)
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pascal[n][k] = (pascal[n-1][k-1] + pascal[n-1][k]) % mod;
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}
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}
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// LATEX_IGNORED_BEGIN pra testar as funções
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int main(){
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int N = 10;
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Pascal(N);
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for(int n=0; n<=N; cout << endl, n++)
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for(int k=0; k<=n; k++)
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cout << pascal[n][k] << "\t";
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cout << endl;
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Combin cb(N);
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for(int n=0; n<=N; cout << endl, n++)
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for(int k=0; k<=n; k++)
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cout << cb.choose(n, k) << "\t";
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cout << endl;
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for(int n=0; n<=N; cout << endl, n++)
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for(int k=0; k<=n; k++)
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cout << cb.chooseLinear(n, k) << "\t";
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cout << endl;
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for(int n=0; n<=N; n++) cout << cb.fat[n] << "\t";
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cout << endl;
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for(int n=0; n<=N; n++) cout << n << "\t";
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cout << endl;
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for(int n=0; n<=N; n++) cout << cb.finv[n] << " ";
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cout << endl;
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}
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// LATEX_IGNORED_END

‎Library/Math/Sieve.cpp

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#include <bits/stdc++.h>
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using namespace std;
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vector<int> calc_prime(int n){ // O(n log n)
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vector<int> prime(n+1, 1);
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for(int i=2; i<=n; i++) if(prime[i] == i)
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for(int j=i+i; j<=n; j+=i)
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prime[j] = false;
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return prime;
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}
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vector<int> calc_phi(int n){ // O(n log n)
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vector<int> phi(n+1);
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for(int i=0; i<=n; i++) phi[i] = i;
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for(int i=2; i<=n; i++) if(phi[i] == i)
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for(int j=i; j<=n; j+=i)
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phi[j] -= phi[j] / i;
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return phi;
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}
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vector<int> calc_mobius(int n){ // O(n log n)
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vector<int> mobius(n+1, 1), prime(n+1, 1);
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for(int i=2, j; i<=n; i++) if(prime[i])
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for(mobius[i]=-1, j=i+i; j<=n; j+=i){
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if((j/i)%i) mobius[j] *= -1;
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else mobius[j] = 0;
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prime[j] = false;
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}
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return mobius;
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}

‎Library/Theorems/Math.tex

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\]
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\item \textbf{Propriedades de Coeficientes Binomiais:}
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\begin{align*}
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\binom{N}{K} = \binom{N}{N - K} &= \frac{N}{K} \binom{N - 1}{K - 1} \\
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\sum_{k=0}^{m} (-1)^k \binom{n}{k} &= (-1)^m \binom{n - 1}{m} \\
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\sum_{m=0}^{n} \binom{m}{k} &= \binom{n+1}{k+1} \\
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\sum_{k=0}^{m} \binom{n + k}{k} &= \binom{n + m + 1}{m} \\
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\sum_{k=0}^{n} \binom{n}{k}^2 &= \binom{2n}{n} \\
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\sum_{k=0}^{n} \binom{n}{k} &= 2^n \\
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\sum_{k=0}^{n} k \binom{n}{k} &= n \cdot 2^{n - 1} \\
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\sum_{k=0}^{n} \binom{n-k}{k} &= F_{n+1}
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\end{align*}
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% \begin{align*}
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% \end{align*}
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% \vspace{-2pt}
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\begin{alignat*}{2}
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\binom{N}{N - K} &= \frac{N}{K} \binom{N - 1}{K - 1} = \binom{N}{K} &\\
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\sum_{k=0}^{m} (-1)^k & \binom{n}{k} = (-1)^m \binom{n - 1}{m} &\\
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\sum_{k=0}^{n} \binom{n}{k} &= 2^n , &
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\sum_{k=0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1} \\
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\sum_{m=0}^{n} \binom{m}{k} &= \binom{n+1}{k+1} , &
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\sum_{k=0}^{n} \binom{n-k}{k} = F_{n+1} \\
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\sum_{k=0}^{m} \binom{n + k}{k} &= \binom{n + m + 1}{m} , &
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\sum_{k=0}^{n} \binom{n}{k}^2 = \binom{2n}{n}
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\end{alignat*}
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\item \textbf{Triângulo de Pascal} \\[0.5ex]
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\includegraphics[width=\linewidth]{math/pascal}
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\item \textbf{Identidades Clássicas:}
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\begin{itemize}

‎pdf/images/math/pascal.jpg

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