Tifa's CP Library

:heavy_check_mark: gen_stirling2_col (src/code/comb/gen_stirling2_col.hpp)

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#ifndef TIFALIBS_COMB_GEN_STIRLING2_COL
#define TIFALIBS_COMB_GEN_STIRLING2_COL

#include "../poly/inv_fps.hpp"
#include "../poly/shl_fps.hpp"
#include "../poly/tsh_fps.hpp"

namespace tifa_libs::math {

// stirling2[i] = {i \\brack k}, i=0,1,...,n
template <class poly>
constexpr poly gen_stirling2_col(u32 n, u32 k, vecu64 const& fact, vecu64 const& ifact) {
  using mint = typename poly::value_type;
  if (k > n) return poly(n + 1);

  auto dfs = [&](auto&& dfs, poly& f, u32 n) -> void {
    if (n == 1) return;
    if (n & 1) {
      dfs(dfs, f, n - 1);
      f.data().push_back(0);
      for (u32 i = n; i; --i) f[i] = f[i - 1] - (n - 1) * f[i];
      f[0] = (-f[0]) * (n - 1);
      return;
    }
    dfs(dfs, f, n / 2);
    f.data().push_back(0);
    f *= tsh_fps(f, mint(mint::mod() - n / 2), fact, ifact);
    f.resize(n + 1);
  };

  poly f{0, 1};
  dfs(dfs, f, k + 1);
  f.resize(k + 2);
  for (u32 i = 0; i < k + 1; ++i) f[i] = f[i + 1];
  f.reverse(k + 1);
  f[k + 1] = 0;
  f.resize(n - k + 1);
  f = inv_fps(f).pre(n + 1);
  return shl_fps(f, k);
}
// stirling2[i] = {i \\brack k}, i=0,1,...,n
template <class poly>
constexpr poly gen_stirling2_col(u32 n, u32 k) {
  using mint = typename poly::value_type;
  return gen_stirling2_col<poly>(n, k, gen_fact(n + 1, mint::mod()), gen_ifact(n + 1, mint::mod()));
}

}  // namespace tifa_libs::math

#endif
#line 1 "src/code/comb/gen_stirling2_col.hpp"



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



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



#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_ifact.hpp"



#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 6 "src/code/comb/gen_ifact.hpp"

namespace tifa_libs::math {

// (i!)^{-1} from i=0..n-1
constexpr vecu64 gen_ifact(u32 n, u64 mod, vecu64 inv) {
  for (u32 i = 2; i < n; ++i) inv[i] = mul_mod_u(inv[i], inv[i - 1], mod);
  return inv;
}
// (i!)^{-1} from i=0..n-1
constexpr vecu64 gen_ifact(u32 n, u64 mod) { return gen_ifact(n, mod, gen_inv(n, mod)); }
// (i!)^{-1} from i=0..n-1
template <class mint>
constexpr vec<mint> gen_ifact(u32 n, vec<mint> inv) {
  for (u32 i = 2; i < n; ++i) inv[i] *= inv[i - 1];
  return inv;
}
// (i!)^{-1} from i=0..n-1
template <class mint>
constexpr vec<mint> gen_ifact(u32 n) { return gen_ifact(n, gen_inv<mint>(n)); }

}  // namespace tifa_libs::math


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

namespace tifa_libs::math {

template <class mint, class ccore>
constexpr poly<mint, ccore> tsh_fps(poly<mint, ccore> const &f, mint c, vecu64 const &fact, vecu64 const &ifact) {
  u32 n = f.size();
  if (n == 1) return f;
  poly<mint, ccore> s = f, p(f.size());
  for (u32 i = 0; i < n; ++i) p[n - i - 1] = f[i] * fact[i];
  {
    mint _ = 1;
    for (u32 i = 0; i < n; ++i, _ *= c) s[i] = _ * ifact[i];
  }
  (p *= s).resize(n);
  for (u32 i = 0; i < n; ++i) s[n - i - 1] = p[i] * ifact[n - i - 1];
  return s;
}
template <class mint, class ccore>
constexpr poly<mint, ccore> tsh_fps(poly<mint, ccore> const &f, mint c) {
  u32 n = f.size();
  if (n == 1) return f;
  return tsh_fps<mint, ccore>(f, c, gen_fact(n, mint::mod()), gen_ifact(n, mint::mod()));
}

}  // namespace tifa_libs::math


#line 7 "src/code/comb/gen_stirling2_col.hpp"

namespace tifa_libs::math {

// stirling2[i] = {i \\brack k}, i=0,1,...,n
template <class poly>
constexpr poly gen_stirling2_col(u32 n, u32 k, vecu64 const& fact, vecu64 const& ifact) {
  using mint = typename poly::value_type;
  if (k > n) return poly(n + 1);

  auto dfs = [&](auto&& dfs, poly& f, u32 n) -> void {
    if (n == 1) return;
    if (n & 1) {
      dfs(dfs, f, n - 1);
      f.data().push_back(0);
      for (u32 i = n; i; --i) f[i] = f[i - 1] - (n - 1) * f[i];
      f[0] = (-f[0]) * (n - 1);
      return;
    }
    dfs(dfs, f, n / 2);
    f.data().push_back(0);
    f *= tsh_fps(f, mint(mint::mod() - n / 2), fact, ifact);
    f.resize(n + 1);
  };

  poly f{0, 1};
  dfs(dfs, f, k + 1);
  f.resize(k + 2);
  for (u32 i = 0; i < k + 1; ++i) f[i] = f[i + 1];
  f.reverse(k + 1);
  f[k + 1] = 0;
  f.resize(n - k + 1);
  f = inv_fps(f).pre(n + 1);
  return shl_fps(f, k);
}
// stirling2[i] = {i \\brack k}, i=0,1,...,n
template <class poly>
constexpr poly gen_stirling2_col(u32 n, u32 k) {
  using mint = typename poly::value_type;
  return gen_stirling2_col<poly>(n, k, gen_fact(n + 1, mint::mod()), gen_ifact(n + 1, mint::mod()));
}

}  // namespace tifa_libs::math


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