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:heavy_check_mark: 二項係数 ($\mathbb{Z}/m\mathbb{Z}$) (src/Math/BinomialCoefficient.hpp)

参考

https://ferin-tech.hatenablog.com/entry/2018/01/17/010829

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Code

#pragma once
#include <vector>
#include "src/NumberTheory/Factors.hpp"
#include "src/Math/mod_inv.hpp"
class BinomialCoefficient {  // mod <= 1e6
 using i64= int64_t;
 struct ModPe {
  ModPe()= default;
  ModPe(int p, int e, std::size_t pre_size= 1 << 14): p(p), e(e), ppows(e + 1, 1) {
   for (int i= 1; i <= e; ++i) ppows[i]= ppows[i - 1] * p;
   for (pp= pe= ppows[e]; std::size_t(pp) * p <= pre_size;) pp*= p;
   q= pp / pe * p, facts.resize(pp, 1);
   for (int qq= 1, l= pp / p; qq < q; qq*= p, l/= p)
    for (int i= 0; i < l; ++i)
     for (int j= i * p + 1; j < i * p + p; ++j) facts[j * qq]= j;
   for (int i= 1; i < pp; ++i) facts[i]= i64(facts[i - 1]) * facts[i] % pe;
   mask= (facts[pp - 1] == pe - 1), ds.resize(q, 0);
   for (int i= 0; i < pp / pe; ++i)
    for (int j= 0, s= ds[i]; j < p; ++j) ds[i * p + j]= s + j;
  }
  int operator()(i64 n, i64 m) const {
   int num= 1, den= 1, x= 0, s= 0;
   if (i64 r= n - m; e > 1)
    for (i64 n1, m1, r1; n > 0; n= n1, m= m1, r= r1) {
     n1= n / pp, m1= m / pp, r1= r / pp;
     num= i64(num) * facts[n - n1 * pp] % pp;
     den= i64(den) * facts[m - m1 * pp] % pp * facts[r - r1 * pp] % pp;
     s+= n1 - m1 - r1, n1= n / q, m1= m / q, r1= r / q;
     x+= ds[m - m1 * q] + ds[r - r1 * q] - ds[n - n1 * q];
    }
   else
    for (i64 n1, m1, r1; n > 0; n= n1, m= m1, r= r1) {
     n1= n / pp, m1= m / pp, r1= r / pp;
     int nr= n - n1 * pp, mr= m - m1 * pp, rr= r - r1 * pp;
     num= i64(num) * facts[nr] % pp;
     den= i64(den) * facts[mr] % pp * facts[rr] % pp;
     s+= n1 - m1 - r1, x+= ds[mr] + ds[rr] - ds[nr];
    }
   if (x >= e * (p - 1)) return 0;
   if (p > 2) x/= p - 1;
   int ret= i64(num) * mod_inv(den, pe) % pe * ppows[x] % pe;
   return (s & mask) && ret > 0 ? pe - ret : ret;
  }
  int p, e, mask, pe, q, pp;
  std::vector<int> ppows, facts, ds;
 };
 int mod;
 std::vector<ModPe> binom_pp;
 std::vector<int> iprods;
public:
 BinomialCoefficient(int mod, std::size_t pre_size= 1 << 14): mod(mod) {
  Factors f(mod);
  if (f.size() == 1) pre_size= 1 << 20;
  int prod= 1;
  for (auto [p, e]: f) {
   binom_pp.emplace_back(ModPe(p, e, pre_size));
   iprods.push_back(mod_inv(prod, binom_pp.back().pe));
   prod*= binom_pp.back().pe;
  }
 }
 inline int nCr(i64 n, i64 r) const {
  assert(r >= 0);
  if (n < r) return 0;
  if (r == 0) return (mod > 1);
  int ret= 0, prod= 1;
  for (size_t i= 0, d, ed= binom_pp.size(); i < ed; ++i, prod*= d) d= binom_pp[i].pe, ret+= i64(binom_pp[i](n, r) + d - ret % d) * iprods[i] % d * prod;
  return ret;
 }
 inline int nHr(i64 n, i64 r) const { return !r ? 1 : nCr(n + r - 1, r); }
};
#line 2 "src/Math/BinomialCoefficient.hpp"
#include <vector>
#line 2 "src/NumberTheory/Factors.hpp"
#include <numeric>
#include <cassert>
#include <iostream>
#include <algorithm>
#line 2 "src/Internal/Remainder.hpp"
namespace math_internal {
using namespace std;
using u8= unsigned char;
using u32= unsigned;
using i64= long long;
using u64= unsigned long long;
using u128= __uint128_t;
struct MP_Na {  // mod < 2^32
 u32 mod;
 constexpr MP_Na(): mod(0) {}
 constexpr MP_Na(u32 m): mod(m) {}
 constexpr inline u32 mul(u32 l, u32 r) const { return u64(l) * r % mod; }
 constexpr inline u32 set(u32 n) const { return n; }
 constexpr inline u32 get(u32 n) const { return n; }
 constexpr inline u32 norm(u32 n) const { return n; }
 constexpr inline u32 plus(u64 l, u32 r) const { return l+= r, l < mod ? l : l - mod; }
 constexpr inline u32 diff(u64 l, u32 r) const { return l-= r, l >> 63 ? l + mod : l; }
};
template <class u_t, class du_t, u8 B> struct MP_Mo {  // mod < 2^32, mod < 2^62
 u_t mod;
 constexpr MP_Mo(): mod(0), iv(0), r2(0) {}
 constexpr MP_Mo(u_t m): mod(m), iv(inv(m)), r2(-du_t(mod) % mod) {}
 constexpr inline u_t mul(u_t l, u_t r) const { return reduce(du_t(l) * r); }
 constexpr inline u_t set(u_t n) const { return mul(n, r2); }
 constexpr inline u_t get(u_t n) const { return n= reduce(n), n >= mod ? n - mod : n; }
 constexpr inline u_t norm(u_t n) const { return n >= mod ? n - mod : n; }
 constexpr inline u_t plus(u_t l, u_t r) const { return l+= r, l < (mod << 1) ? l : l - (mod << 1); }
 constexpr inline u_t diff(u_t l, u_t r) const { return l-= r, l >> (B - 1) ? l + (mod << 1) : l; }
private:
 u_t iv, r2;
 static constexpr u_t inv(u_t n, int e= 6, u_t x= 1) { return e ? inv(n, e - 1, x * (2 - x * n)) : x; }
 constexpr inline u_t reduce(const du_t &w) const { return u_t(w >> B) + mod - ((du_t(u_t(w) * iv) * mod) >> B); }
};
using MP_Mo32= MP_Mo<u32, u64, 32>;
using MP_Mo64= MP_Mo<u64, u128, 64>;
struct MP_Br {  // 2^20 < mod <= 2^41
 u64 mod;
 constexpr MP_Br(): mod(0), x(0) {}
 constexpr MP_Br(u64 m): mod(m), x((u128(1) << 84) / m) {}
 constexpr inline u64 mul(u64 l, u64 r) const { return rem(u128(l) * r); }
 static constexpr inline u64 set(u64 n) { return n; }
 constexpr inline u64 get(u64 n) const { return n >= mod ? n - mod : n; }
 constexpr inline u64 norm(u64 n) const { return n >= mod ? n - mod : n; }
 constexpr inline u64 plus(u64 l, u64 r) const { return l+= r, l < (mod << 1) ? l : l - (mod << 1); }
 constexpr inline u64 diff(u64 l, u64 r) const { return l-= r, l >> 63 ? l + (mod << 1) : l; }
private:
 u64 x;
 constexpr inline u128 quo(const u128 &n) const { return (n * x) >> 84; }
 constexpr inline u64 rem(const u128 &n) const { return n - quo(n) * mod; }
};
template <class du_t, u8 B> struct MP_D2B1 {  // mod < 2^63, mod < 2^64
 u64 mod;
 constexpr MP_D2B1(): mod(0), s(0), d(0), v(0) {}
 constexpr MP_D2B1(u64 m): mod(m), s(__builtin_clzll(m)), d(m << s), v(u128(-1) / d) {}
 constexpr inline u64 mul(u64 l, u64 r) const { return rem((u128(l) * r) << s) >> s; }
 constexpr inline u64 set(u64 n) const { return n; }
 constexpr inline u64 get(u64 n) const { return n; }
 constexpr inline u64 norm(u64 n) const { return n; }
 constexpr inline u64 plus(du_t l, u64 r) const { return l+= r, l < mod ? l : l - mod; }
 constexpr inline u64 diff(du_t l, u64 r) const { return l-= r, l >> B ? l + mod : l; }
private:
 u8 s;
 u64 d, v;
 constexpr inline u64 rem(const u128 &u) const {
  u128 q= (u >> 64) * v + u;
  u64 r= u64(u) - (q >> 64) * d - d;
  if (r > u64(q)) r+= d;
  if (r >= d) r-= d;
  return r;
 }
};
using MP_D2B1_1= MP_D2B1<u64, 63>;
using MP_D2B1_2= MP_D2B1<u128, 127>;
template <class u_t, class MP> constexpr u_t pow(u_t x, u64 k, const MP &md) {
 for (u_t ret= md.set(1);; x= md.mul(x, x))
  if (k & 1 ? ret= md.mul(ret, x) : 0; !(k>>= 1)) return ret;
}
}
#line 3 "src/NumberTheory/is_prime.hpp"
namespace math_internal {
template <class Uint, class MP, u32... args> constexpr bool miller_rabin(Uint n) {
 const MP md(n);
 const Uint s= __builtin_ctzll(n - 1), d= n >> s, one= md.set(1), n1= md.norm(md.set(n - 1));
 for (u32 a: (u32[]){args...})
  if (Uint b= a % n; b)
   if (Uint p= md.norm(pow(md.set(b), d, md)); p != one)
    for (int i= s; p != n1; p= md.norm(md.mul(p, p)))
     if (!(--i)) return 0;
 return 1;
}
}
constexpr bool is_prime(unsigned long long n) {
 if (n < 2 || n % 6 % 4 != 1) return (n | 1) == 3;
 if (n < (1 << 30)) return math_internal::miller_rabin<unsigned, math_internal::MP_Mo32, 2, 7, 61>(n);
 if (n < (1ull << 62)) return math_internal::miller_rabin<unsigned long long, math_internal::MP_Mo64, 2, 325, 9375, 28178, 450775, 9780504, 1795265022>(n);
 if (n < (1ull << 63)) return math_internal::miller_rabin<unsigned long long, math_internal::MP_D2B1_1, 2, 325, 9375, 28178, 450775, 9780504, 1795265022>(n);
 return math_internal::miller_rabin<unsigned long long, math_internal::MP_D2B1_2, 2, 325, 9375, 28178, 450775, 9780504, 1795265022>(n);
}
#line 2 "src/Math/binary_gcd.hpp"
#include <type_traits>
#line 4 "src/Math/binary_gcd.hpp"
#include <cstdint>
template <class Int> constexpr int bsf(Int a) {
 if constexpr (sizeof(Int) == 16) {
  uint64_t lo= a & uint64_t(-1);
  return lo ? __builtin_ctzll(lo) : 64 + __builtin_ctzll(a >> 64);
 } else if constexpr (sizeof(Int) == 8) return __builtin_ctzll(a);
 else return __builtin_ctz(a);
}
template <class Int> constexpr Int binary_gcd(Int a, Int b) {
 if (a == 0 || b == 0) return a + b;
 int n= bsf(a), m= bsf(b), s= 0;
 for (a>>= n, b>>= m; a != b;) {
  Int d= a - b;
  bool f= a > b;
  s= bsf(d), b= f ? b : a, a= (f ? d : -d) >> s;
 }
 return a << std::min(n, m);
}
#line 9 "src/NumberTheory/Factors.hpp"
namespace math_internal {
template <class T> constexpr void bubble_sort(T *bg, T *ed) {
 for (int sz= ed - bg, i= 0; i < sz; i++)
  for (int j= sz; --j > i;)
   if (auto tmp= bg[j - 1]; bg[j - 1] > bg[j]) bg[j - 1]= bg[j], bg[j]= tmp;
}
template <class T, size_t _Nm> struct ConstexprArray {
 constexpr size_t size() const { return sz; }
 constexpr auto &operator[](int i) const { return dat[i]; }
 constexpr auto *begin() const { return dat; }
 constexpr auto *end() const { return dat + sz; }
protected:
 T dat[_Nm]= {};
 size_t sz= 0;
 friend ostream &operator<<(ostream &os, const ConstexprArray &r) {
  os << "[";
  for (size_t i= 0; i < r.sz; ++i) os << r[i] << ",]"[i == r.sz - 1];
  return os;
 }
};
class Factors: public ConstexprArray<pair<u64, uint16_t>, 16> {
 template <class Uint, class MP> static constexpr Uint rho(Uint n, Uint c) {
  const MP md(n);
  auto f= [&md, c](Uint x) { return md.plus(md.mul(x, x), c); };
  const Uint m= 1LL << (__lg(n) / 5);
  Uint x= 1, y= md.set(2), z= 1, q= md.set(1), g= 1;
  for (Uint r= 1, i= 0; g == 1; r<<= 1) {
   for (x= y, i= r; i--;) y= f(y);
   for (Uint k= 0; k < r && g == 1; g= binary_gcd<Uint>(md.get(q), n), k+= m)
    for (z= y, i= min(m, r - k); i--;) y= f(y), q= md.mul(q, md.diff(y, x));
  }
  if (g == n) do {
    z= f(z), g= binary_gcd<Uint>(md.get(md.diff(z, x)), n);
   } while (g == 1);
  return g;
 }
 static constexpr u64 find_prime_factor(u64 n) {
  if (is_prime(n)) return n;
  for (u64 i= 100; i--;)
   if (n= n < (1 << 30) ? rho<u32, MP_Mo32>(n, i + 1) : n < (1ull << 62) ? rho<u64, MP_Mo64>(n, i + 1) : n < (1ull << 62) ? rho<u64, MP_D2B1_1>(n, i + 1) : rho<u64, MP_D2B1_2>(n, i + 1); is_prime(n)) return n;
  return 0;
 }
 constexpr void init(u64 n) {
  for (u64 p= 2; p < 98 && p * p <= n; ++p)
   if (n % p == 0)
    for (dat[sz++].first= p; n % p == 0;) n/= p, ++dat[sz - 1].second;
  for (u64 p= 0; n > 1; dat[sz++].first= p)
   for (p= find_prime_factor(n); n % p == 0;) n/= p, ++dat[sz].second;
 }
public:
 constexpr Factors()= default;
 constexpr Factors(u64 n) { init(n), bubble_sort(dat, dat + sz); }
};
}
using math_internal::Factors;
constexpr uint64_t totient(const Factors &f) {
 uint64_t ret= 1, i= 0;
 for (auto [p, e]: f)
  for (ret*= p - 1, i= e; --i;) ret*= p;
 return ret;
}
constexpr auto totient(uint64_t n) { return totient(Factors(n)); }
template <class Uint= uint64_t> std::vector<Uint> enumerate_divisors(const Factors &f) {
 int k= 1;
 for (auto [p, e]: f) k*= e + 1;
 std::vector<Uint> ret(k, 1);
 k= 1;
 for (auto [p, e]: f) {
  int sz= k;
  for (Uint pw= 1; pw*= p, e--;)
   for (int j= 0; j < sz;) ret[k++]= ret[j++] * pw;
 }
 return ret;
}
template <class Uint> std::vector<Uint> enumerate_divisors(Uint n) { return enumerate_divisors<Uint>(Factors(n)); }
#line 2 "src/Math/mod_inv.hpp"
#include <utility>
#line 5 "src/Math/mod_inv.hpp"
template <class Uint> constexpr inline Uint mod_inv(Uint a, Uint mod) {
 std::make_signed_t<Uint> x= 1, y= 0, z= 0;
 for (Uint q= 0, b= mod, c= 0; b;) z= x, x= y, y= z - y * (q= a / b), c= a, a= b, b= c - b * q;
 return assert(a == 1), x < 0 ? mod - (-x) % mod : x % mod;
}
#line 5 "src/Math/BinomialCoefficient.hpp"
class BinomialCoefficient {  // mod <= 1e6
 using i64= int64_t;
 struct ModPe {
  ModPe()= default;
  ModPe(int p, int e, std::size_t pre_size= 1 << 14): p(p), e(e), ppows(e + 1, 1) {
   for (int i= 1; i <= e; ++i) ppows[i]= ppows[i - 1] * p;
   for (pp= pe= ppows[e]; std::size_t(pp) * p <= pre_size;) pp*= p;
   q= pp / pe * p, facts.resize(pp, 1);
   for (int qq= 1, l= pp / p; qq < q; qq*= p, l/= p)
    for (int i= 0; i < l; ++i)
     for (int j= i * p + 1; j < i * p + p; ++j) facts[j * qq]= j;
   for (int i= 1; i < pp; ++i) facts[i]= i64(facts[i - 1]) * facts[i] % pe;
   mask= (facts[pp - 1] == pe - 1), ds.resize(q, 0);
   for (int i= 0; i < pp / pe; ++i)
    for (int j= 0, s= ds[i]; j < p; ++j) ds[i * p + j]= s + j;
  }
  int operator()(i64 n, i64 m) const {
   int num= 1, den= 1, x= 0, s= 0;
   if (i64 r= n - m; e > 1)
    for (i64 n1, m1, r1; n > 0; n= n1, m= m1, r= r1) {
     n1= n / pp, m1= m / pp, r1= r / pp;
     num= i64(num) * facts[n - n1 * pp] % pp;
     den= i64(den) * facts[m - m1 * pp] % pp * facts[r - r1 * pp] % pp;
     s+= n1 - m1 - r1, n1= n / q, m1= m / q, r1= r / q;
     x+= ds[m - m1 * q] + ds[r - r1 * q] - ds[n - n1 * q];
    }
   else
    for (i64 n1, m1, r1; n > 0; n= n1, m= m1, r= r1) {
     n1= n / pp, m1= m / pp, r1= r / pp;
     int nr= n - n1 * pp, mr= m - m1 * pp, rr= r - r1 * pp;
     num= i64(num) * facts[nr] % pp;
     den= i64(den) * facts[mr] % pp * facts[rr] % pp;
     s+= n1 - m1 - r1, x+= ds[mr] + ds[rr] - ds[nr];
    }
   if (x >= e * (p - 1)) return 0;
   if (p > 2) x/= p - 1;
   int ret= i64(num) * mod_inv(den, pe) % pe * ppows[x] % pe;
   return (s & mask) && ret > 0 ? pe - ret : ret;
  }
  int p, e, mask, pe, q, pp;
  std::vector<int> ppows, facts, ds;
 };
 int mod;
 std::vector<ModPe> binom_pp;
 std::vector<int> iprods;
public:
 BinomialCoefficient(int mod, std::size_t pre_size= 1 << 14): mod(mod) {
  Factors f(mod);
  if (f.size() == 1) pre_size= 1 << 20;
  int prod= 1;
  for (auto [p, e]: f) {
   binom_pp.emplace_back(ModPe(p, e, pre_size));
   iprods.push_back(mod_inv(prod, binom_pp.back().pe));
   prod*= binom_pp.back().pe;
  }
 }
 inline int nCr(i64 n, i64 r) const {
  assert(r >= 0);
  if (n < r) return 0;
  if (r == 0) return (mod > 1);
  int ret= 0, prod= 1;
  for (size_t i= 0, d, ed= binom_pp.size(); i < ed; ++i, prod*= d) d= binom_pp[i].pe, ret+= i64(binom_pp[i](n, r) + d - ret % d) * iprods[i] % d * prod;
  return ret;
 }
 inline int nHr(i64 n, i64 r) const { return !r ? 1 : nCr(n + r - 1, r); }
};
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