Tifa's CP Library

:heavy_check_mark: gen_stirling1_col (src/code/comb/gen_stirling1_col.hpp)

Depends on

Verified with

Code

#ifndef TIFALIBS_COMB_GEN_STIRLING1_COL
#define TIFALIBS_COMB_GEN_STIRLING1_COL

#include "../poly/pow_fps.hpp"
#include "gen_fact.hpp"
#include "gen_inv.hpp"
#include "gen_invseq.hpp"

namespace tifa_libs::math {

// stirling1[i] = {i \\brack k}, i=0,1,...,n
template <class poly, bool with_sgn = true>
constexpr poly gen_stirling1_col(u32 n, u32 k, vecu64 const& fact, vecu64 const& inv, vecu64 const& invfact) {
  if (n < k) return poly(n + 1);
  poly f(n + 1);
  for (u32 i = 1; i <= n; ++i) f[i] = inv[i];
  f = pow_fps(f, k) * invfact[k];
  for (u32 i = k; i <= n; ++i) f[i] *= fact[i];
  if constexpr (with_sgn)
    for (u32 i = k ^ 1; i <= n; i += 2) f[i] = -f[i];
  return f;
}
// stirling1[i] = {i \\brack k}, i=0,1,...,n
template <class poly, bool with_sgn = true>
constexpr poly gen_stirling1_col(u32 n, u32 k) {
  using mint = typename poly::value_type;
  auto fact = gen_fact(n + 1, mint::mod());
  return gen_stirling1_col<poly, with_sgn>(n, k, fact, gen_inv(n + 1, mint::mod()), gen_invseq(fact, mint::mod()));
}

}  // namespace tifa_libs::math

#endif
#line 1 "src/code/comb/gen_stirling1_col.hpp"



#line 1 "src/code/poly/pow_fps.hpp"



#line 1 "src/code/math/qpow.hpp"



#line 1 "src/code/util/util.hpp"



#include <bits/stdc++.h>

template <class T>
constexpr T abs(T x) { return x < 0 ? -x : x; }

using i8 = int8_t;
using i16 = int16_t;
using i32 = int32_t;
using i64 = int64_t;
using i128 = __int128_t;
using isz = ptrdiff_t;

using u8 = uint8_t;
using u16 = uint16_t;
using u32 = uint32_t;
using u64 = uint64_t;
using u128 = __uint128_t;
using usz = size_t;

using f32 = float;
using f64 = double;
using f128 = long double;

template <class T>
using ptt = std::pair<T, T>;
template <class T>
using pt3 = std::tuple<T, T, T>;
template <class T>
using pt4 = std::tuple<T, T, T, T>;

template <class T, usz N>
using arr = std::array<T, N>;
template <class T>
using vec = std::vector<T>;
template <class T>
using vvec = vec<vec<T>>;
template <class T>
using v3ec = vec<vvec<T>>;
template <class U, class T>
using vecp = vec<std::pair<U, T>>;
template <class U, class T>
using vvecp = vvec<std::pair<U, T>>;
template <class T>
using vecpt = vec<ptt<T>>;
template <class T>
using vvecpt = vvec<ptt<T>>;

template <class T, class C = std::less<T>>
using pq = std::priority_queue<T, vec<T>, C>;
template <class T>
using pqg = std::priority_queue<T, vec<T>, std::greater<T>>;

using strn = std::string;
using strnv = std::string_view;

using vecu = vec<u32>;
using vvecu = vvec<u32>;
using v3ecu = v3ec<u32>;
using vecu64 = vec<u64>;
using vecb = vec<bool>;
using vvecb = vvec<bool>;

#ifdef ONLINE_JUDGE
#undef assert
#define assert(x) 42
#endif

using namespace std::literals;

constexpr i8 operator""_i8(unsigned long long x) { return (i8)x; }
constexpr i16 operator""_i16(unsigned long long x) { return (i16)x; }
constexpr i32 operator""_i32(unsigned long long x) { return (i32)x; }
constexpr i64 operator""_i64(unsigned long long x) { return (i64)x; }
constexpr isz operator""_iz(unsigned long long x) { return (isz)x; }

constexpr u8 operator""_u8(unsigned long long x) { return (u8)x; }
constexpr u16 operator""_u16(unsigned long long x) { return (u16)x; }
constexpr u32 operator""_u32(unsigned long long x) { return (u32)x; }
constexpr u64 operator""_u64(unsigned long long x) { return (u64)x; }
constexpr usz operator""_uz(unsigned long long x) { return (usz)x; }

inline const auto fn_0 = [](auto&&...) {};


#line 5 "src/code/math/qpow.hpp"

namespace tifa_libs::math {

template <class T>
constexpr T qpow(T a, u64 b, T const& init_v = T{1}) {
  T res = init_v;
  for (; b; b >>= 1, a = a * a)
    if (b & 1) res = res * a;
  return res;
}

}  // namespace tifa_libs::math


#line 1 "src/code/poly/exp_fps.hpp"



#line 1 "src/code/poly/ln_fps.hpp"



#line 1 "src/code/poly/deriv_fps.hpp"



#line 1 "src/code/poly/poly.hpp"



#line 5 "src/code/poly/poly.hpp"

namespace tifa_libs::math {

// clang-format off
enum ccore_t { ct_FFT, ct_3NTT, ct_NTT, ct_CNTT };
// clang-format on

template <class mint, class ccore>
requires requires(ccore cc, vec<mint> l, vec<mint> const &r, u32 sz) {
  { ccore::ct_cat } -> std::same_as<ccore_t const &>;
  cc.conv(l, r);
  cc.conv(l, r, sz);
}
class poly {
  vec<mint> d;

 public:
  using value_type = mint;
  using data_type = vec<value_type>;
  using ccore_type = ccore;
  static inline ccore_type conv_core;

  explicit constexpr poly(u32 sz = 1, value_type const &val = value_type{}) : d(sz, val) {}
  constexpr poly(typename data_type::const_iterator begin, typename data_type::const_iterator end) : d(begin, end) {}
  constexpr poly(std::initializer_list<value_type> v) : d(v) {}
  template <class T>
  explicit constexpr poly(vec<T> const &v) : d(v) {}

  friend constexpr std::istream &operator>>(std::istream &is, poly &poly) {
    for (auto &val : poly.d) is >> val;
    return is;
  }
  friend constexpr std::ostream &operator<<(std::ostream &os, poly const &poly) {
    if (!poly.size()) return os;
    for (u32 i = 1; i < poly.size(); ++i) os << poly[i - 1] << ' ';
    return os << poly.d.back();
  }

  constexpr u32 size() const { return (u32)d.size(); }
  constexpr bool empty() const {
    for (auto &&i : d)
      if (i != 0) return 0;
    return 1;
  }
  constexpr data_type &data() { return d; }
  constexpr data_type const &data() const { return d; }

  constexpr value_type &operator[](u32 x) { return d[x]; }
  constexpr value_type const &operator[](u32 x) const { return d[x]; }
  constexpr value_type operator()(value_type x) const {
    value_type ans = 0;
    for (u32 i = size() - 1; ~i; --i) ans = ans * x + d[i];
    return ans;
  }

  template <class F>
  requires requires(F f, u32 idx, mint &val) {
    f(idx, val);
  }
  constexpr void apply_range(u32 l, u32 r, F &&f) {
    assert(l < r && r <= size());
    for (u32 i = l; i < r; ++i) f(i, d[i]);
  }
  template <class F>
  constexpr void apply(F &&f) { apply_range(0, size(), std::forward<F>(f)); }
  constexpr void resize(u32 size) { d.resize(size); }
  constexpr poly pre(u32 size) const {
    poly _ = *this;
    _.resize(size);
    return _;
  }
  constexpr void strip() {
    auto it = std::find_if(d.rbegin(), d.rend(), [](auto const &x) { return x != 0; });
    d.resize(usz(d.rend() - it));
    if (d.empty()) d.push_back(value_type(0));
  }
  friend poly stripped(poly p) {
    p.strip();
    return p;
  }
  constexpr void reverse(u32 n = 0) { std::reverse(d.begin(), d.begin() + (n ? n : size())); }
  constexpr void conv(poly const &r, u32 ans_size = 0) { conv_core.conv(d, r.d, ans_size); }

  constexpr poly operator-() const {
    poly ret = *this;
    ret.apply([](u32, auto &v) { v = -v; });
    return ret;
  }

  friend constexpr poly operator+(poly p, value_type c) {
    p[0] += c;
    return p;
  }
  friend constexpr poly operator+(value_type c, poly const &p) { return p + c; }
  friend constexpr poly operator-(poly p, value_type c) {
    p[0] -= c;
    return p;
  }
  friend constexpr poly operator-(value_type c, poly const &p) { return p - c; }

  constexpr poly &operator*=(value_type c) {
    apply([&c](u32, auto &v) { v *= c; });
    return *this;
  }
  friend constexpr poly operator*(poly p, value_type c) { return p *= c; }
  friend constexpr poly operator*(value_type c, poly p) { return p *= c; }

  constexpr poly &operator+=(poly const &r) {
    if (!r.size()) return *this;
    resize(std::max(size(), r.size()));
    apply_range(0, r.size(), [&r](u32 i, auto &v) { v += r[i]; });
    return *this;
  }
  friend constexpr poly operator+(poly l, poly const &r) { return l += r; }

  constexpr poly &operator-=(poly const &r) {
    if (!r.size()) return *this;
    resize(std::max(size(), r.size()));
    apply_range(0, r.size(), [&r](u32 i, auto &v) { v -= r[i]; });
    return *this;
  }
  friend constexpr poly operator-(poly l, poly const &r) { return l -= r; }

  constexpr poly &operator*=(poly const &r) {
    if (!r.size()) {
      resize(1);
      d[0] = 0;
      return *this;
    }
    conv(r);
    return *this;
  }
  friend constexpr poly operator*(poly l, poly const &r) { return l *= r; }

  constexpr auto operator<=>(poly const &r) const { return stripped(*this).d <=> stripped(r).d; }
  constexpr bool operator==(poly const &r) const { return stripped(*this).d == stripped(r).d; }
};

}  // namespace tifa_libs::math


#line 5 "src/code/poly/deriv_fps.hpp"

namespace tifa_libs::math {

template <class mint, class ccore>
constexpr poly<mint, ccore> deriv_fps(poly<mint, ccore> const &p) {
  auto _ = p;
  for (u32 i = 1; i < _.size(); ++i) _[i - 1] = _[i] * i;
  _.data().back() = 0;
  return _;
}

}  // namespace tifa_libs::math


#line 1 "src/code/poly/int_fps.hpp"



#line 5 "src/code/poly/int_fps.hpp"

namespace tifa_libs::math {

template <class mint, class ccore>
constexpr poly<mint, ccore> int_fps(poly<mint, ccore> const &p) {
  auto _ = p;
  for (u32 i = _.size() - 1; i; --i) _[i] = _[i - 1] * mint(i).inv();
  _[0] = 0;
  return _;
}

}  // namespace tifa_libs::math


#line 1 "src/code/poly/inv_fps.hpp"



#line 5 "src/code/poly/inv_fps.hpp"

namespace tifa_libs::math {

template <class mint, class ccore>
constexpr poly<mint, ccore> inv_fps(poly<mint, ccore> const &p, u32 n = 0) {
  assert(p[0] != 0);
  if (!n) n = p.size();
  poly<mint, ccore> a{p[0].inv()};
  for (u32 i = 1; i < n; i *= 2) a = (a * 2 - (a * a * p).pre(i * 2)).pre(i * 2);
  return a.pre(n);
}

}  // namespace tifa_libs::math


#line 7 "src/code/poly/ln_fps.hpp"

namespace tifa_libs::math {

template <class mint, class ccore>
constexpr poly<mint, ccore> ln_fps(poly<mint, ccore> const &p, u32 n = 0) {
  assert(p[0] == 1);
  if (!n) n = p.size();
  auto _ = deriv_fps(p).pre(n);
  _.conv(inv_fps(p, n));
  return int_fps(_).pre(n);
}

}  // namespace tifa_libs::math


#line 5 "src/code/poly/exp_fps.hpp"

namespace tifa_libs::math {

template <class mint, class ccore>
constexpr poly<mint, ccore> exp_fps(poly<mint, ccore> p, u32 n = 0) {
  assert(p[0] == 0);
  if (!n) n = p.size();
  p[0] += 1;
  poly<mint, ccore> a{1};
  for (u32 i = 1; i < n; i *= 2) a = (a * (p.pre(i * 2) - ln_fps(a, i * 2))).pre(i * 2);
  return a.pre(n);
}

}  // namespace tifa_libs::math


#line 1 "src/code/poly/shl_fps.hpp"



#line 5 "src/code/poly/shl_fps.hpp"

namespace tifa_libs::math {

template <class mint, class ccore>
constexpr poly<mint, ccore> shl_fps(poly<mint, ccore> const &p, usz x) {
  if (!x) return p;
  auto _ = p;
  if (x >= _.size()) {
    std::ranges::fill(_.data(), 0);
    return _;
  }
  std::fill(_.data().begin(), std::move_backward(_.data().begin(), std::prev(_.data().end(), (isz)x), _.data().end()), 0);
  return _;
}

}  // namespace tifa_libs::math


#line 1 "src/code/poly/shr_fps.hpp"



#line 5 "src/code/poly/shr_fps.hpp"

namespace tifa_libs::math {

template <class mint, class ccore>
constexpr poly<mint, ccore> shr_fps(poly<mint, ccore> const &p, usz x) {
  if (!x) return p;
  auto _ = p;
  if (x >= p.size()) {
    std::ranges::fill(_.data(), 0);
    return _;
  }
  std::fill(std::move(_.data().begin() + (isz)x, _.data().end(), _.data().begin()), _.data().end(), 0);
  return _;
}
template <class mint, class ccore>
constexpr poly<mint, ccore> shr_strip_fps(poly<mint, ccore> const &p, usz x) {
  auto _ = shr_fps(p, x);
  _.strip();
  return _;
}

}  // namespace tifa_libs::math


#line 9 "src/code/poly/pow_fps.hpp"

namespace tifa_libs::math {

template <class mint, class ccore>
constexpr poly<mint, ccore> pow_fps(poly<mint, ccore> const &p, u64 y, u32 n = 0) {
  if (!n) n = p.size();
  if (y == 0) {
    poly<mint, ccore> _(n);
    if (n) _[0] = 1;
    return _;
  }
  if (y == 1) return p;
  u32 l0 = u32(std::ranges::find_if(p.data(), [](auto const &x) { return x != 0; }) - p.data().begin());
  if ((u128)l0 * y >= n) return poly<mint, ccore>(n);
  if (l0) {
    auto _ = shr_fps(p, l0), g = pow_fps(_, y, u32(n - l0 * y));
    g.resize(n);
    return shl_fps(g, l0 * y);
  }
  auto _ = p;
  mint _0 = p[0];
  if (_0 != 1) _ *= _0.inv();
  _ = exp_fps(ln_fps(_) * y);
  if (_0 != 1) _ *= qpow(_0, y);
  _.resize(n);
  return _;
}

}  // namespace tifa_libs::math


#line 1 "src/code/comb/gen_fact.hpp"



#line 1 "src/code/math/mul_mod_u.hpp"



#line 5 "src/code/math/mul_mod_u.hpp"

namespace tifa_libs::math {

constexpr u64 mul_mod_u(u64 a, u64 b, u64 mod) {
  if (std::bit_width(a) + std::bit_width(b) <= 64) return a * b % mod;
  else return (u64)((u128)a * b % mod);
}

}  // namespace tifa_libs::math


#line 5 "src/code/comb/gen_fact.hpp"

namespace tifa_libs::math {

// i! from i=0..n-1
constexpr vecu64 gen_fact(u32 n, u64 mod) {
  if (n == 0) return {};
  if (n == 1) return {1};
  vecu64 ans(n);
  ans[0] = ans[1] = 1;
  for (u32 i = 2; i < n; ++i) ans[i] = mul_mod_u(ans[i - 1], i, mod);
  return ans;
}
// i! from i=0..n-1
template <class mint>
constexpr vec<mint> gen_fact(u32 n) {
  vec<mint> ans(n);
  auto _ = gen_fact(n, mint::mod());
  for (u32 i = 0; i < n; ++i) ans[i] = _[i];
  return ans;
}

}  // namespace tifa_libs::math


#line 1 "src/code/comb/gen_inv.hpp"



#line 5 "src/code/comb/gen_inv.hpp"

namespace tifa_libs::math {

// i^{-1} from i=0..n-1
constexpr vecu64 gen_inv(u32 n, u64 mod) {
  if (n == 0) return {};
  if (n == 1) return {1};
  vecu64 ans(n);
  ans[0] = ans[1] = 1;
  for (u32 i = 2; i < n; ++i) ans[i] = mul_mod_u(mod - mod / i, ans[mod % i], mod);
  return ans;
}
// i^{-1} from i=0..n-1
template <class mint>
constexpr vec<mint> gen_inv(u32 n) {
  vec<mint> ans(n);
  auto _ = gen_inv(n, mint::mod());
  for (u32 i = 0; i < n; ++i) ans[i] = _[i];
  return ans;
}

}  // namespace tifa_libs::math


#line 1 "src/code/comb/gen_invseq.hpp"



#line 1 "src/code/nt/inverse.hpp"



#line 1 "src/code/nt/inv_gcd.hpp"



#line 1 "src/code/math/safe_mod.hpp"



#line 1 "src/code/util/traits.hpp"



#line 5 "src/code/util/traits.hpp"

namespace tifa_libs {

template <class T>
concept iterable_c = requires(T v) {
  { v.begin() } -> std::same_as<typename T::iterator>;
  { v.end() } -> std::same_as<typename T::iterator>;
};

template <class T>
concept container_c = iterable_c<T> && !std::derived_from<T, std::basic_string<typename T::value_type>>;

template <class T>
constexpr bool is_char_v = std::is_same_v<T, char> || std::is_same_v<T, signed char> || std::is_same_v<T, unsigned char>;
template <class T>
concept char_c = is_char_v<T>;

template <class T>
constexpr bool is_s128_v = std::is_same_v<T, __int128_t> || std::is_same_v<T, __int128>;
template <class T>
concept s128_c = is_s128_v<T>;

template <class T>
constexpr bool is_u128_v = std::is_same_v<T, __uint128_t> || std::is_same_v<T, unsigned __int128>;
template <class T>
concept u128_c = is_u128_v<T>;

template <class T>
constexpr bool is_i128_v = is_s128_v<T> || is_u128_v<T>;
template <class T>
concept i128_c = is_u128_v<T>;

template <class T>
constexpr bool is_int_v = std::is_integral_v<T> || is_i128_v<T>;
template <class T>
concept int_c = is_int_v<T>;

template <class T>
constexpr bool is_sint_v = is_s128_v<T> || (is_int_v<T> && std::is_signed_v<T>);
template <class T>
concept sint_c = is_sint_v<T>;

template <class T>
constexpr bool is_uint_v = is_u128_v<T> || (is_int_v<T> && std::is_unsigned_v<T>);
template <class T>
concept uint_c = is_uint_v<T>;

template <class T>
concept mint_c = requires(T x) {
  { x.mod() } -> uint_c;
  { x.val() } -> uint_c;
};

template <class T>
constexpr bool is_arithm_v = std::is_arithmetic_v<T> || is_int_v<T>;
template <class T>
concept arithm_c = is_arithm_v<T>;

template <class T>
struct to_sint : std::make_signed<T> {};
template <>
struct to_sint<u128> {
  using type = u128;
};
template <>
struct to_sint<i128> {
  using type = u128;
};
template <class T>
using to_sint_t = typename to_sint<T>::type;

template <class T>
struct to_uint : std::make_unsigned<T> {};
template <>
struct to_uint<u128> {
  using type = u128;
};
template <>
struct to_uint<i128> {
  using type = u128;
};
template <class T>
using to_uint_t = typename to_uint<T>::type;

}  // namespace tifa_libs


#line 5 "src/code/math/safe_mod.hpp"

namespace tifa_libs::math {

template <sint_c T>
constexpr T safe_mod(T x, to_uint_t<T> mod) { return ((x %= (T)mod) < 0 ? x + (T)mod : x); }

}  // namespace tifa_libs::math


#line 1 "src/code/nt/exgcd.hpp"



#line 5 "src/code/nt/exgcd.hpp"

namespace tifa_libs::math {

// @return tuple(g, x, y) s.t. g = gcd(a, b), xa + yb = g, |x| + |y| is the minimal (primary) and x <= y (secondarily)
template <sint_c T>
constexpr auto exgcd(T a, T b) {
  T x1 = 1, x2 = 0, x3 = 0, x4 = 1;
  while (b) {
    T c = a / b;
    std::tie(x1, x2, x3, x4, a, b) = std::make_tuple(x3, x4, x1 - x3 * c, x2 - x4 * c, b, a - b * c);
  }
  return std::make_tuple(to_uint_t<T>(a), x1, x2);
}

}  // namespace tifa_libs::math


#line 6 "src/code/nt/inv_gcd.hpp"

namespace tifa_libs::math {

template <uint_c T>
constexpr ptt<T> inv_gcd(T n, T mod) {
  using U = to_sint_t<T>;
  auto [g, x, y] = exgcd(U(n % mod), (U)mod);
  return {g, safe_mod(x, mod)};
}

}  // namespace tifa_libs::math


#line 5 "src/code/nt/inverse.hpp"

namespace tifa_libs::math {

template <uint_c T, uint_c U>
constexpr U inverse(T n, U mod) {
  auto [g, x] = inv_gcd(U(n % mod), mod);
  assert(g == 1);
  return x;
}

}  // namespace tifa_libs::math


#line 6 "src/code/comb/gen_invseq.hpp"

namespace tifa_libs::math {

// i^{-1} for i in v
constexpr vecu64 gen_invseq(vecu64 const &v, u64 mod) {
  u32 n = (u32)v.size();
  if (n == 0) return {};
  if (n == 1) return {inverse(v[0], mod)};
  vecu64 ans(n);
  ans[0] = v[1];
  for (u32 i = 1; i < n; ++i) ans[i] = mul_mod_u(ans[i - 1], v[i], mod);
  u64 _ = inverse(ans.back(), mod);
  for (u32 i = n - 1; i; --i) {
    ans[i] = mul_mod_u(_, ans[i - 1], mod);
    _ = mul_mod_u(_, v[i], mod);
  }
  ans[0] = _;
  return ans;
}
// i^{-1} for i in v
template <class mint>
constexpr vec<mint> gen_invseq(vec<mint> const &v) {
  u32 n = (u32)v.size();
  vec<mint> ans(n);
  auto _ = gen_invseq(v, mint::mod());
  for (u32 i = 0; i < n; ++i) ans[i] = _[i];
  return ans;
}

}  // namespace tifa_libs::math


#line 8 "src/code/comb/gen_stirling1_col.hpp"

namespace tifa_libs::math {

// stirling1[i] = {i \\brack k}, i=0,1,...,n
template <class poly, bool with_sgn = true>
constexpr poly gen_stirling1_col(u32 n, u32 k, vecu64 const& fact, vecu64 const& inv, vecu64 const& invfact) {
  if (n < k) return poly(n + 1);
  poly f(n + 1);
  for (u32 i = 1; i <= n; ++i) f[i] = inv[i];
  f = pow_fps(f, k) * invfact[k];
  for (u32 i = k; i <= n; ++i) f[i] *= fact[i];
  if constexpr (with_sgn)
    for (u32 i = k ^ 1; i <= n; i += 2) f[i] = -f[i];
  return f;
}
// stirling1[i] = {i \\brack k}, i=0,1,...,n
template <class poly, bool with_sgn = true>
constexpr poly gen_stirling1_col(u32 n, u32 k) {
  using mint = typename poly::value_type;
  auto fact = gen_fact(n + 1, mint::mod());
  return gen_stirling1_col<poly, with_sgn>(n, k, fact, gen_inv(n + 1, mint::mod()), gen_invseq(fact, mint::mod()));
}

}  // namespace tifa_libs::math


Back to top page