スターリング数 ($\mathbb{F}_p$) (src/Math/StirlingNumber.hpp)

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• Last update: 2024-10-01 17:03:34+09:00
• Include: #include "src/Math/StirlingNumber.hpp"

計算量

前処理 $O(p^2)$, クエリ $O(\log n)$

参考

https://maspypy.com/stirling-数を-p-で割った余りの計算

Depends on

• 剰余の高速化 (src/Internal/Remainder.hpp)
• 素数判定 (src/NumberTheory/is_prime.hpp)

Verified with

• test/yosupo/stirling_1_small_p_large_n.test.cpp
• test/yosupo/stirling_2_small_p_large_n.test.cpp

Code

#pragma once
#include <vector>
#include <algorithm>
#include <cassert>
#include <cstdint>
#include "src/NumberTheory/is_prime.hpp"
class StirlingNumber {
 const uint16_t p;
 std::vector<std::vector<uint16_t>> c, s1, s2;
 void buildS1() {
  s1.resize(p), s1[0]= {1};
  for (int i= 1, j, t; i < p; s1[i][i]= 1, i++)
   for (s1[i].resize(i + 1, 0), j= 1, t= p - i + 1; j < i; j++) s1[i][j]= (t * s1[i - 1][j] + s1[i - 1][j - 1]) % p;
 }
 void buildS2() {
  s2.resize(p), s2[0]= {1};
  for (int i= 1, j; i < p; s2[i][i]= 1, i++)
   for (s2[i].resize(i + 1, 0), j= 1; j < i; j++) s2[i][j]= (j * s2[i - 1][j] + s2[i - 1][j - 1]) % p;
 }
public:
 StirlingNumber(uint32_t p_, bool first= true, bool second= true): p(p_), c(p) {
  assert(is_prime(p_)), assert(p_ < (1 << 15)), c[0]= {1};
  for (int i= 1, j; i < p; i++)
   for (c[i]= c[i - 1], c[i].emplace_back(0), j= 1; j <= i; j++) c[i][j]-= p & -((c[i][j]+= c[i - 1][j - 1]) >= p);
  if (first) buildS1();
  if (second) buildS2();
 }
 int nCk(uint64_t n, uint64_t k) {
  if (k > n) return 0;
  int ret= 1, i, j;
  for (k= std::min(k, n - k); k; ret= ret * c[i][j] % p, n/= p, k/= p)
   if (i= n % p, j= k % p; j > i) return 0;
  return ret;
 }
 int S1(uint64_t n, uint64_t k) {
  if (k > n) return 0;
  uint64_t i= n / p;
  if (i > k) return 0;
  int64_t a= (k - i) / (p - 1);
  uint16_t j= n % p, b= (k - i) % (p - 1);
  if (!b && j) b+= (p - 1), a-= 1;
  if (a < 0 || i < a || b > j) return 0;
  return (j= nCk(i, a) * s1[j][b] % p) && ((i + a) & 1) ? p - j : j;
 }
 int S2(uint64_t n, uint64_t k) {
  if (k > n) return 0;
  if (!n) return 1;
  uint64_t i= k / p;
  if (n <= i) return 0;
  uint64_t a= (n - i - 1) / (p - 1);
  uint16_t j= k % p, b= (n - i) - a * (p - 1);
  if (j > b) return 0;
  return b == p - 1 && !j ? nCk(a, i - 1) : nCk(a, i) * s2[b][j] % p;
 }
};

#line 2 "src/Math/StirlingNumber.hpp"
#include <vector>
#include <algorithm>
#include <cassert>
#include <cstdint>
#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 7 "src/Math/StirlingNumber.hpp"
class StirlingNumber {
 const uint16_t p;
 std::vector<std::vector<uint16_t>> c, s1, s2;
 void buildS1() {
  s1.resize(p), s1[0]= {1};
  for (int i= 1, j, t; i < p; s1[i][i]= 1, i++)
   for (s1[i].resize(i + 1, 0), j= 1, t= p - i + 1; j < i; j++) s1[i][j]= (t * s1[i - 1][j] + s1[i - 1][j - 1]) % p;
 }
 void buildS2() {
  s2.resize(p), s2[0]= {1};
  for (int i= 1, j; i < p; s2[i][i]= 1, i++)
   for (s2[i].resize(i + 1, 0), j= 1; j < i; j++) s2[i][j]= (j * s2[i - 1][j] + s2[i - 1][j - 1]) % p;
 }
public:
 StirlingNumber(uint32_t p_, bool first= true, bool second= true): p(p_), c(p) {
  assert(is_prime(p_)), assert(p_ < (1 << 15)), c[0]= {1};
  for (int i= 1, j; i < p; i++)
   for (c[i]= c[i - 1], c[i].emplace_back(0), j= 1; j <= i; j++) c[i][j]-= p & -((c[i][j]+= c[i - 1][j - 1]) >= p);
  if (first) buildS1();
  if (second) buildS2();
 }
 int nCk(uint64_t n, uint64_t k) {
  if (k > n) return 0;
  int ret= 1, i, j;
  for (k= std::min(k, n - k); k; ret= ret * c[i][j] % p, n/= p, k/= p)
   if (i= n % p, j= k % p; j > i) return 0;
  return ret;
 }
 int S1(uint64_t n, uint64_t k) {
  if (k > n) return 0;
  uint64_t i= n / p;
  if (i > k) return 0;
  int64_t a= (k - i) / (p - 1);
  uint16_t j= n % p, b= (k - i) % (p - 1);
  if (!b && j) b+= (p - 1), a-= 1;
  if (a < 0 || i < a || b > j) return 0;
  return (j= nCk(i, a) * s1[j][b] % p) && ((i + a) & 1) ? p - j : j;
 }
 int S2(uint64_t n, uint64_t k) {
  if (k > n) return 0;
  if (!n) return 1;
  uint64_t i= k / p;
  if (n <= i) return 0;
  uint64_t a= (n - i - 1) / (p - 1);
  uint16_t j= k % p, b= (n - i) - a * (p - 1);
  if (j > b) return 0;
  return b == p - 1 && !j ? nCk(a, i - 1) : nCk(a, i) * s2[b][j] % p;
 }
};

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