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‎docs/fps/famous-sequences.md‎

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## 分割数
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非負整数 $n$ の分割数 $p_n$ とは，$n$ をいくつかの正整数の和（順序を区別しない）で表す方法の数．
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母関数は $\prod_{k=1}^{\infty}\frac{1}{1-x^k}$ である．
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オイラーの五角数定理より
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$$\prod_{k=1}^{\infty}(1-x^k)=\sum_{k=-\infty}^{\infty}(-1)^kx^{k(3k-1)/2}$$
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であることを用いれば $O(N\log N)$ 時間で $p_0,p_1,\dots,p_N$ が列挙できる．
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## 第一種スターリング数
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$s(n,k)$ を以下で定める．
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$$x(x-1)\cdots(x-(n-1))=\sum_{k=0}^{n}s(n,k)x^k$$
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$s(n,i)$ の $0\leq i\leq n$ での列挙が $O(n\log n)$ 時間でできる．
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- 分割統治をするが，一方の結果を Taylor Shift すればもう一方の結果を得られる．
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また
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$$\sum_{i,j}s(i,j)\frac{x^i}{i!}y^j=(1+x)^y=\exp(y\log(1+x))=\sum_{j}(\log(1+x))^j\frac{y^j}{j!}$$
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となるので $s(i,k)$ の $k\leq i\leq n$ での列挙が $O((n-k)\log (n-k))$ 時間でできる．
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## 第二種スターリング数
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$S(n,k)$ を以下で定める．
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$$x^n=\sum_{k=0}^{n}S(n,k)x(x-1)\cdots(x-(k-1))$$
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整数 $i$ について $x=i$ としたとき
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$$i^n=i![y^i]\left(\sum_{k=0}^{n}S(n,k)y^k\right)e^y$$
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と表示できるので
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$$\sum_{k=0}^{n}S(n,k)y^k=e^{-y}\left(\sum_{i=0}^{\infty}\frac{i^n}{i!}y^i\right)$$
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であり，$S(n,i)$ の $0\leq i\leq n$ での列挙が $O(n\log n)$ 時間でできる．
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また
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$$\sum_{i=0}^{\infty}\frac{i^n}{i!}y^i=n![x^n]\sum_{i=0}^{\infty}\frac{e^{ix}}{i!}y^i=n![x^n]\exp(e^xy)$$
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より
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$$\sum_{n,k}S(n,k)\frac{x^n}{n!}y^k=\exp((e^x-1)y)=\sum_{i\geq 0}(e^x-1)^i\frac{y^i}{i!}$$
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となるので $S(i,k)$ の $k\leq i\leq n$ での列挙が $O((n-k)\log (n-k))$ 時間でできる．

‎fps/famous-sequences.hpp‎

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#pragma once
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#include "modint/factorial.hpp"
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#include "modint/power-table.hpp"
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#include "fps/formal-power-series.hpp"
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#include "fps/taylor-shift.hpp"
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template <class mint>
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FormalPowerSeries<mint> PartitionFunction(int n) {
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FormalPowerSeries<mint> g(n + 1);
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for (int k = 0; k * (3 * k - 1) / 2 <= n; k++) g[k * (3 * k - 1) / 2] += k & 1 ? -1 : 1;
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for (int k = 1; k * (3 * k + 1) / 2 <= n; k++) g[k * (3 * k + 1) / 2] += k & 1 ? -1 : 1;
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return g.inv(n + 1);
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}
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template <class mint>
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FormalPowerSeries<mint> FirstKindStirlingNumbers(int n) {
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FormalPowerSeries<mint> f{1};
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for (int l = 30; l >= 0; l--) {
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if (f.size() > 1) f *= TaylorShift(f, mint(-(n >> (l + 1))));
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if ((n >> l) & 1) f = (f << 1) - f * mint((n >> l) - 1);
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}
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return f;
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}
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template <class mint>
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FormalPowerSeries<mint> FirstKindStirlingNumbersFixedK(int n, int k) {
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using fact = Factorial<mint>;
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if (k > n) return FormalPowerSeries<mint>{};
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FormalPowerSeries<mint> f(n - k + 1);
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for (int i = 0; i < f.size(); i++) f[i] = fact::inv(i + 1) * (i & 1 ? -1 : 1);
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f = f.pow(k);
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f *= fact::fact_inv(k);
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for (int i = 0; i < f.size(); i++) f[i] *= fact::fact(i + k);
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return f;
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}
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template <class mint>
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FormalPowerSeries<mint> SecondKindStirlingNumbers(int n) {
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using fact = Factorial<mint>;
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FormalPowerSeries<mint> f(n + 1);
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for (int i = 0; i < f.size(); i++) f[i] = fact::fact_inv(i) * (i & 1 ? -1 : 1);
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FormalPowerSeries<mint> g(PowerTable<mint>(n, n));
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for (int i = 0; i < g.size(); i++) g[i] *= fact::fact_inv(i);
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f *= g;
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f.resize(n + 1);
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return f;
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}
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template <class mint>
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FormalPowerSeries<mint> SecondKindStirlingNumbersFixedK(int n, int k) {
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using fact = Factorial<mint>;
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if (k > n) return FormalPowerSeries<mint>{};
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FormalPowerSeries<mint> f(n - k + 1);
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for (int i = 0; i < f.size(); i++) f[i] = fact::fact_inv(i + 1);
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f = f.pow(k);
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f *= fact::fact_inv(k);
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for (int i = 0; i < f.size(); i++) f[i] *= fact::fact(i + k);
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return f;
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}
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/**
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* @brief 有名数列
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* @docs docs/fps/famous-sequences.md
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*/

‎template/debug.hpp‎

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@@ -11,7 +11,7 @@ void _show(int i, T name) {
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cerr << '\n';
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}
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template <class T1, class T2, class... T3>
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void _show(int i, const T1 &a, const T2 &b, const T3 &...c) {
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void _show(int i, const T1& a, const T2& b, const T3&... c) {
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for (; a[i] != ',' && a[i] != '\0'; i++) cerr << a[i];
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cerr << ":" << b << " ";
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_show(i + 1, a, c...);

‎template/inout.hpp‎

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} fast;
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template <class T1, class T2>
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istream &operator>>(istream &is, pair<T1, T2> &p) {
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istream& operator>>(istream& is, pair<T1, T2>& p) {
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return is >> p.first >> p.second;
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}
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template <class T1, class T2>
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ostream &operator<<(ostream &os, const pair<T1, T2> &p) {
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ostream& operator<<(ostream& os, const pair<T1, T2>& p) {
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return os << p.first << " " << p.second;
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}
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template <class T>
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istream &operator>>(istream &is, vector<T> &a) {
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for (auto &v : a) is >> v;
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istream& operator>>(istream& is, vector<T>& a) {
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for (auto& v : a) is >> v;
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return is;
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}
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template <class T>
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ostream &operator<<(ostream &os, const vector<T> &a) {
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ostream& operator<<(ostream& os, const vector<T>& a) {
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for (auto it = a.begin(); it != a.end();) {
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os << *it;
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if (++it != a.end()) os << " ";
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}
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return os;
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}
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template <class T>
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ostream &operator<<(ostream &os, const set<T> &st) {
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ostream& operator<<(ostream& os, const set<T>& st) {
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os << "{";
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for (auto it = st.begin(); it != st.end();) {
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os << *it;
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return os;
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}
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template <class T1, class T2>
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ostream &operator<<(ostream &os, const map<T1, T2> &mp) {
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ostream& operator<<(ostream& os, const map<T1, T2>& mp) {
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os << "{";
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for (auto it = mp.begin(); it != mp.end();) {
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os << it->first << ":" << it->second;
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void in() {}
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template <typename T, class... U>
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void in(T &t, U &...u) {
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void in(T& t, U&... u) {
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cin >> t;
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in(u...);
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}
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void out() { cout << "\n"; }
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template <typename T, class... U, char sep = ' '>
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void out(const T &t, const U &...u) {
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void out(const T& t, const U&... u) {
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cout << t;
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if (sizeof...(u)) cout << sep;
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out(u...);

‎template/util.hpp‎

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using u128 = __uint128_t;
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template <class T, class S = T>
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S SUM(const vector<T> &a) {
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S SUM(const vector<T>& a) {
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return accumulate(ALL(a), S(0));
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}
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template <class T>
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inline bool chmin(T &a, T b) {
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inline bool chmin(T& a, T b) {
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if (a > b) {
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a = b;
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return true;
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}
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return false;
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}
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template <class T>
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inline bool chmax(T &a, T b) {
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inline bool chmax(T& a, T b) {
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if (a < b) {
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a = b;
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return true;

‎verify/fps/LC_partition_function.test.cpp‎

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#define PROBLEM "https://judge.yosupo.jp/problem/partition_function"
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#include "template/template.hpp"
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#include "modint/modint.hpp"
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using mint = ModInt<998244353>;
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#include "fps/fps-ntt-friendly.hpp"
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using fps = FormalPowerSeries<mint>;
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#include "fps/famous-sequences.hpp"
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int main() {
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int n;
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in(n);
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out(PartitionFunction<mint>(n));
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}

‎verify/fps/LC_stirling_number_of_the_first_kind.test.cpp‎

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#define PROBLEM "https://judge.yosupo.jp/problem/stirling_number_of_the_first_kind"
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#include "template/template.hpp"
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#include "modint/modint.hpp"
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using mint = ModInt<998244353>;
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#include "fps/fps-ntt-friendly.hpp"
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using fps = FormalPowerSeries<mint>;
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#include "fps/famous-sequences.hpp"
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int main() {
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int n;
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in(n);
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out(FirstKindStirlingNumbers<mint>(n));
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}

‎verify/fps/LC_stirling_number_of_the_first_kind_fixed_k.test.cpp‎

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#define PROBLEM "https://judge.yosupo.jp/problem/stirling_number_of_the_first_kind_fixed_k"
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#include "template/template.hpp"
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#include "modint/modint.hpp"
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using mint = ModInt<998244353>;
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#include "fps/fps-ntt-friendly.hpp"
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using fps = FormalPowerSeries<mint>;
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#include "fps/famous-sequences.hpp"
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int main() {
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int n, k;
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in(n, k);
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out(FirstKindStirlingNumbersFixedK<mint>(n, k));
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}

‎verify/fps/LC_stirling_number_of_the_second_kind.test.cpp‎

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#define PROBLEM "https://judge.yosupo.jp/problem/stirling_number_of_the_second_kind"
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#include "template/template.hpp"
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#include "modint/modint.hpp"
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using mint = ModInt<998244353>;
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#include "fps/fps-ntt-friendly.hpp"
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using fps = FormalPowerSeries<mint>;
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#include "fps/famous-sequences.hpp"
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int main() {
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int n;
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in(n);
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out(SecondKindStirlingNumbers<mint>(n));
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}

‎verify/fps/LC_stirling_number_of_the_second_kind_fixed_k.test.cpp‎

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#define PROBLEM "https://judge.yosupo.jp/problem/stirling_number_of_the_second_kind_fixed_k"
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#include "template/template.hpp"
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#include "modint/modint.hpp"
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using mint = ModInt<998244353>;
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#include "fps/fps-ntt-friendly.hpp"
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using fps = FormalPowerSeries<mint>;
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#include "fps/famous-sequences.hpp"
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int main() {
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int n, k;
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in(n, k);
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out(SecondKindStirlingNumbersFixedK<mint>(n, k));
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}

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