题解 - [Luogu P5824] 十二重计数法

发表于 2023-05-01 更新于 2024-07-02 分类于算法竞赛，题解本文字数： 1.1k 阅读时长 ≈ 4 分钟

题目链接

原始题面

题目背景

组合数学是一门古老而迷人的学科.

传说早在 \(114514\) 年前，一位名为忆哀的神灵来到地球，发现了人类 —— 另一种有智慧的物种.

她觉得这很有趣，为了加速人类文明的发展，她向人间传下了一类计数问题 —— 十二重计数，这也正是组合数学的开端.

而只有搞明白这类问题，才能在组合数学上继续深入.

题目描述

有 \(n\) 个球和 \(m\) 个盒子，要全部装进盒子里.
还有一些限制条件，那么有多少种方法放球？（与放的先后顺序无关）

限制条件分别如下:

\(\text{I}\): 球之间互不相同，盒子之间互不相同.
\(\text{II}\): 球之间互不相同，盒子之间互不相同，每个盒子至多装一个球.
\(\text{III}\): 球之间互不相同，盒子之间互不相同，每个盒子至少装一个球.

\(\text{IV}\): 球之间互不相同，盒子全部相同.
\(\text{V}\): 球之间互不相同，盒子全部相同，每个盒子至多装一个球.
\(\text{VI}\): 球之间互不相同，盒子全部相同，每个盒子至少装一个球.

\(\text{VII}\): 球全部相同，盒子之间互不相同.
\(\text{VIII}\): 球全部相同，盒子之间互不相同，每个盒子至多装一个球.
\(\text{IX}\): 球全部相同，盒子之间互不相同，每个盒子至少装一个球.

\(\text{X}\): 球全部相同，盒子全部相同.
\(\text{XI}\): 球全部相同，盒子全部相同，每个盒子至多装一个球.
\(\text{XII}\): 球全部相同，盒子全部相同，每个盒子至少装一个球.

由于答案可能很大，所以要对 \(998244353\) 取模.

输入格式

仅一行两个正整数 \(n,m\)

输出格式

输出十二行，每行一个整数，对应每一种限制条件的答案.

样例 #1

样例输入 #1

13 6

样例输出 #1

提示

【数据范围】
对于 \(100\%\) 的数据，\(1\le n,m \le 2\times 10^5\).

orz \(\mathsf E \color{red}\mathsf{ntropyIncreaser}\)

解题思路

Ⅰ

不难发现答案即为 \(m^n\)

Ⅱ

不难发现答案即为 \(m^{\underline{n}}\)

Ⅲ

考虑二项式反演

设

\(f(n,m)\) 为 \(n\) 个球和 \(m\) 个盒子，球之间互不相同，盒子之间互不相同，每个盒子至少装一个球的方案数
\(g(n,m)\) 为 \(n\) 个球和 \(m\) 个盒子，球之间互不相同，盒子之间互不相同的方案数

则

\[ \begin{aligned} &m^n=g(n,m)=\sum_{i=0}^m\binom{m}{i}f(n,i)\\ \implies &f(n,m)=\sum_{i=0}^m\binom{m}{i}(-1)^{m-i}g(n,i)=\sum_{i=0}^m\binom{m}{i}(-1)^{m-i}i^n \end{aligned} \]

Ⅳ

考虑第二类 Stirling 数，不难发现答案为

\[ \sum_{i=0}^m{n \brace i} \]

使用卷积可以快速求出答案，具体而言，令

\[ f(x)=\sum_{i=0}^{\infty}\frac{i^n}{i!}x^i \]
\[ g(x)=\sum_{i=0}^{\infty}\frac{(-1)^i}{i!}x^i \]

则

\[ h(x)=f(x)g(x)=\sum_{i=0}^{\infty}x^i\sum_{j=0}^i\frac{(-1)^{i-j}j^n}{j!(i-j)!}=\sum_{i=0}^{\infty}{n\brace i}x^i \]

Ⅴ

不难发现当 \(m<n\) 时答案为 \(0\), 否则为 \(1\)

Ⅵ

考虑第二类 Stirling 数，不难发现答案为 \({n \brace m}\)

可以和 Ⅳ 一起求

Ⅶ

考虑隔板法，不难发现答案为 \(\binom{n+m-1}{m-1}\)

Ⅷ

不难发现答案即为 \(\binom{m}{n}\)

Ⅸ

考虑隔板法，不难发现答案为 \(\binom{n-1}{m-1}\)

Ⅹ

设 \(f(n,m)\) 为所求，则

\[ f(n,m)=f(n,m-1)+f(n-m,m) \]

考虑 OGF

\[ \begin{aligned} &\begin{aligned} F_m(x)&=\sum_{i=0}^{\infty}f(i,m)x^i\\ &=F_{m-1}(x)+x^m F_m(x) \end{aligned}\\ \implies& F_m(x)=\frac{1}{1-x^m}F_{m-1}(x)\\ \implies& F_m(x)=\prod_{i=1}^{\infty}\frac{1}{1-x^i}\\ \implies& \ln F_m(x)=-\sum_{i=1}^{\infty}\ln(1-x^i)\\ \implies& \ln F_m(x)=\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} \frac{x^{ij}}{j}\\ \implies& F_m(x)=\exp\left(\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} \frac{x^{ij}}{j}\right) \end{aligned} \]

Ⅺ

同 Ⅴ

Ⅻ

类似 Ⅹ, 答案为 \(f(n-m,m)\)

代码参考

Show code

Luogu_P5824view raw

/*
 * @Author: Tifa
 * @Description: From <https://github.com/Tiphereth-A/CP-archives>
 * !!! ATTENEION: All the context below is licensed under a 
 *  GNU Affero General Public License, Version 3.
 *  See <https://www.gnu.org/licenses/agpl-3.0.txt>.
 */
#include <bits/stdc++.h>
#include <bits/stdtr1c++.h>
#include <ext/algorithm>
#include <ext/rb_tree>
#include <ext/rope>
#include <ext/vstring.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/priority_queue.hpp>
#include <ext/pb_ds/exception.hpp>
#include <ext/pb_ds/hash_policy.hpp>
#include <ext/pb_ds/list_update_policy.hpp>
#include <ext/pb_ds/tree_policy.hpp>
#include <ext/pb_ds/trie_policy.hpp>
namespace NdVector {
template <size_t N, class Tp>
struct ndvector: public std::vector<ndvector<N - 1, Tp>> {
  static_assert(N > 0, "N should be positive");
  using base_tp = ndvector<N - 1, Tp>;
  using base = std::vector<base_tp>;
  using self = ndvector<N, Tp>;
  template <class T, typename... Ts>
  ndvector(T &&n, Ts &&...args): base(n, base_tp(args...)) {}
  constexpr size_t dim() const { return N; }
  template <class T>
  void fill(T &&x) {
    for (auto &i : *this) i.fill(x);
  }
};
template <class Tp>
struct ndvector<1, Tp>: public std::vector<Tp> {
  using base = std::vector<Tp>;
  using self = ndvector<1, Tp>;
  template <class T>
  ndvector(T &&n): base(n) {}
  constexpr size_t dim() const { return 1; }
  template <class T>
  void fill(T &&x) {
    std::fill(this->begin(), this->end(), x);
  }
};
}  // namespace NdVector
struct CustomHash {
  static constexpr uint64_t splitmix64(uint64_t x) {
    x += 0x9e3779b97f4a7c15;
    x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
    x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
    return x ^ (x >> 31);
  }
  static constexpr size_t append(size_t x, size_t y) {
    return x ^ (y >> 1) ^ ((y & 1) << (sizeof(size_t) * 8 - 1));
  }
  size_t operator()(uint64_t x) const {
    static const uint64_t FIXED_RANDOM =
      std::chrono::steady_clock::now().time_since_epoch().count();
    return splitmix64(x + FIXED_RANDOM);
  }
  template <class Tp, class Up>
  size_t operator()(std::pair<Tp, Up> const &p) const {
    return append((*this)(p.first), (*this)(p.second));
  }
  template <typename... Ts>
  size_t operator()(std::tuple<Ts...> const &tp) const {
    size_t ret = 0;
    std::apply(
      [&](Ts const &...targs) { ((ret = append(ret, (*this)(targs))), ...); },
      tp);
    return ret;
  }
  template <
    class Tp,
    std::enable_if_t<std::is_same<decltype(std::declval<Tp>().begin()),
                                  typename Tp::iterator>::value &&
                     std::is_same<decltype(std::declval<Tp>().end()),
                                  typename Tp::iterator>::value> * = nullptr>
  size_t operator()(Tp const &tp) const {
    size_t ret = 0;
    for (auto &&i : tp) ret = append(ret, (*this)(i));
    return ret;
  }
};
using u32 = uint32_t;
using i64 = int64_t;
#define for_(i, l, r, vars...)                                            \
  for (std::make_signed_t<decltype(l + r)> i = (l), i##end = (r), ##vars; \
       i <= i##end;                                                       \
       ++i)
#define rfor_(i, r, l, vars...)                                           \
  for (std::make_signed_t<decltype(r - l)> i = (r), i##end = (l), ##vars; \
       i >= i##end;                                                       \
       --i)
template <class Tp>
constexpr auto chkmin(Tp &a, Tp b) -> bool {
  return b < a ? a = b, true : false;
}
template <class Tp>
constexpr auto chkmax(Tp &a, Tp b) -> bool {
  return a < b ? a = b, true : false;
}
template <class Tp>
constexpr auto ispow2(Tp i) -> bool {
  return i && (i & -i) == i;
}
#define TPL_SIZE_(Tuple) std::tuple_size_v<std::remove_reference_t<Tuple>>
namespace tuple_detail_ {
template <std::size_t Begin, class Tuple, std::size_t... Is>
constexpr auto subtuple_impl_(Tuple &&t, std::index_sequence<Is...>) {
  return std::make_tuple(std::get<Is + Begin>(t)...);
}
template <class Tuple, class BinOp, std::size_t... Is>
constexpr auto
apply2_impl_(BinOp &&f, Tuple &&lhs, Tuple &&rhs, std::index_sequence<Is...>) {
  return std::make_tuple(
    std::forward<BinOp>(f)(std::get<Is>(lhs), std::get<Is>(rhs))...);
}
}  // namespace tuple_detail_
template <std::size_t Begin, std::size_t Len, class Tuple>
constexpr auto subtuple(Tuple &&t) {
  static_assert(Begin <= TPL_SIZE_(Tuple) && Len <= TPL_SIZE_(Tuple) &&
                  Begin + Len <= TPL_SIZE_(Tuple),
                "Out of range");
  return tuple_detail_::subtuple_impl_<Begin>(t,
                                              std::make_index_sequence<Len>());
}
template <std::size_t Pos, class Tp, class Tuple>
constexpr auto tuple_push(Tp &&v, Tuple &&t) {
  static_assert(TPL_SIZE_(Tuple) > 0, "Pop from empty tuple");
  return std::tuple_cat(subtuple<0, Pos>(t),
                        std::make_tuple(v),
                        subtuple<Pos, TPL_SIZE_(Tuple) - Pos>(t));
}
template <class Tp, class Tuple>
constexpr auto tuple_push_front(Tp &&v, Tuple &&t) {
  return tuple_push<0>(v, t);
}
template <class Tp, class Tuple>
constexpr auto tuple_push_back(Tp &&v, Tuple &&t) {
  return tuple_push<TPL_SIZE_(Tuple)>(v, t);
}
template <std::size_t Pos, class Tuple>
constexpr auto tuple_pop(Tuple &&t) {
  static_assert(TPL_SIZE_(Tuple) > 0, "Pop from empty tuple");
  return std::tuple_cat(subtuple<0, Pos>(t),
                        subtuple<Pos + 1, TPL_SIZE_(Tuple) - Pos - 1>(t));
}
template <class Tuple>
constexpr auto tuple_pop_front(Tuple &&t) {
  return tuple_pop<0>(t);
}
template <class Tuple>
constexpr auto tuple_pop_back(Tuple &&t) {
  return tuple_pop<TPL_SIZE_(Tuple) - 1>(t);
}
template <class Tuple, class BinOp>
constexpr auto apply2(BinOp &&f, Tuple &&lhs, Tuple &&rhs) {
  return tuple_detail_::apply2_impl_(
    f, lhs, rhs, std::make_index_sequence<TPL_SIZE_(Tuple)>());
}
#define OO_PTEQ_(op)                                                          \
  template <class Tp, class Up>                                               \
  constexpr auto operator op(std::pair<Tp, Up> lhs,                           \
                             const std::pair<Tp, Up> &rhs) {                  \
    return std::pair<Tp, Up>{lhs.first op rhs.first,                          \
                             lhs.second op rhs.second};                       \
  }                                                                           \
  template <class... Ts>                                                      \
  constexpr auto operator op(std::tuple<Ts...> const &lhs,                    \
                             std::tuple<Ts...> const &rhs) {                  \
    return apply2([](auto &&l, auto &&r) { return l op r; }, lhs, rhs);       \
  }                                                                           \
  template <class Tp, class Up>                                               \
  constexpr std::pair<Tp, Up> &operator op##=(std::pair<Tp, Up> &lhs,         \
                                              const std::pair<Tp, Up> &rhs) { \
    lhs.first op## = rhs.first;                                               \
    lhs.second op## = rhs.second;                                             \
    return lhs;                                                               \
  }                                                                           \
  template <class... Ts>                                                      \
  constexpr auto operator op##=(std::tuple<Ts...> &lhs,                       \
                                const std::tuple<Ts...> &rhs) {               \
    return lhs = lhs op rhs;                                                  \
  }
OO_PTEQ_(+)
OO_PTEQ_(-)
OO_PTEQ_(*)
OO_PTEQ_(/)
OO_PTEQ_(%)
OO_PTEQ_(&)
OO_PTEQ_(|)
OO_PTEQ_(^)
OO_PTEQ_(<<)
OO_PTEQ_(>>)
#undef OO_PTEQ_
#undef TPL_SIZE_
template <class Tp, class Up>
std::istream &operator>>(std::istream &is, std::pair<Tp, Up> &p) {
  return is >> p.first >> p.second;
}
template <class Tp, class Up>
std::ostream &operator<<(std::ostream &os, const std::pair<Tp, Up> &p) {
  return os << p.first << ' ' << p.second;
}
template <typename... Ts>
std::istream &operator>>(std::istream &is, std::tuple<Ts...> &p) {
  std::apply([&](Ts &...targs) { ((is >> targs), ...); }, p);
  return is;
}
template <typename... Ts>
std::ostream &operator<<(std::ostream &os, const std::tuple<Ts...> &p) {
  std::apply(
    [&](Ts const &...targs) {
      std::size_t n{0};
      ((os << targs << (++n != sizeof...(Ts) ? " " : "")), ...);
    },
    p);
  return os;
}
template <
  class Ch,
  class Tr,
  class Ct,
  std::enable_if_t<std::is_same<decltype(std::declval<Ct>().begin()),
                                typename Ct::iterator>::value &&
                   std::is_same<decltype(std::declval<Ct>().end()),
                                typename Ct::iterator>::value> * = nullptr>
std::basic_ostream<Ch, Tr> &operator<<(std::basic_ostream<Ch, Tr> &os,
                                       const Ct &x) {
  if (x.begin() == x.end()) return os;
  for (auto it = x.begin(); it != x.end() - 1; ++it) os << *it << ' ';
  os << x.back();
  return os;
}
const u32 OFFSET = 5;
const u32 N = 4e5 + OFFSET;
const u32 M = 2e5 + OFFSET;
const i64 MOD = 1e9 + 7;
using namespace std;
namespace Common {
namespace MODINT {
constexpr int64_t safe_mod(int64_t x, int64_t m) {
  return (x %= m) < 0 ? x + m : x;
}
constexpr std::pair<int64_t, int64_t> invgcd(int64_t a, int64_t b) {
  if ((a = safe_mod(a, b)) == 0) return {b, 0};
  int64_t s = b, m0 = 0;
  for (int64_t q = 0, _ = 0, m1 = 1; a;) {
    _ = s - a * (q = s / a);
    s = a;
    a = _;
    _ = m0 - m1 * q;
    m0 = m1;
    m1 = _;
  }
  return {s, m0 + (m0 < 0 ? b / s : 0)};
}
template <uint32_t MOD>
class Mint {
  static_assert(MOD >= 1);
  using self = Mint<MOD>;

protected:
  uint32_t v_;

public:
  static constexpr uint32_t mod() { return MOD; }
  static constexpr self raw(uint32_t v) {
    self x;
    x.v_ = v;
    return x;
  }
  constexpr Mint(): v_(0) {}
  template <class T,
            std::enable_if_t<std::is_integral<T>::value &&
                             std::is_signed<T>::value> * = nullptr>
  constexpr Mint(T v): Mint() {
    int64_t x = (int64_t)(v % (int64_t)mod());
    v_ = (uint32_t)(x + (x < 0 ? mod() : 0));
  }
  template <class T,
            std::enable_if_t<std::is_integral<T>::value &&
                             std::is_unsigned<T>::value> * = nullptr>
  constexpr Mint(T v): v_((uint32_t)(v % mod())) {}
  friend std::istream &operator>>(std::istream &is, self &x) {
    int64_t xx;
    is >> xx;
    xx %= mod();
    x.v_ = (uint32_t)(xx + (xx < 0 ? mod() : 0));
    return is;
  }
  friend std::ostream &operator<<(std::ostream &os, const self &x) {
    return os << x.v_;
  }
  constexpr const uint32_t &val() const { return v_; }
  constexpr explicit operator uint32_t() const { return val(); }
  constexpr uint32_t &data() { return v_; }
  constexpr self &operator++() {
    if (++v_ == mod()) v_ = 0;
    return *this;
  }
  constexpr self &operator--() {
    if (!v_) v_ = mod();
    --v_;
    return *this;
  }
  constexpr self operator++(int) {
    self result = *this;
    ++*this;
    return result;
  }
  constexpr self operator--(int) {
    self result = *this;
    --*this;
    return result;
  }
  constexpr self &operator+=(const self &rhs) {
    v_ += rhs.v_;
    if (v_ >= mod()) v_ -= mod();
    return *this;
  }
  constexpr self &operator-=(const self &rhs) {
    v_ -= rhs.v_;
    if (v_ >= mod()) v_ += mod();
    return *this;
  }
  constexpr self &operator*=(const self &rhs) {
    v_ = (uint32_t)((uint64_t)v_ * rhs.v_ % mod());
    return *this;
  }
  constexpr self &operator/=(const self &rhs) {
    return *this = *this * inverse(rhs);
  }
  constexpr self operator+() const { return *this; }
  constexpr self operator-() const { return self() - *this; }
  constexpr friend self pow(self x, uint64_t y) {
    self res(1);
    for (; y; y >>= 1, x *= x)
      if (y & 1) res *= x;
    return res;
  }
  constexpr friend self inverse(const self &x) {
    auto &&_ = invgcd(x.v_, self::mod());
    if (_.first != 1) throw std::runtime_error("Inverse not exist");
    return _.second;
  }
  constexpr friend self operator+(self lhs, const self &rhs) {
    return lhs += rhs;
  }
  constexpr friend self operator-(self lhs, const self &rhs) {
    return lhs -= rhs;
  }
  constexpr friend self operator*(self lhs, const self &rhs) {
    return lhs *= rhs;
  }
  constexpr friend self operator/(self lhs, const self &rhs) {
    return lhs /= rhs;
  }
  constexpr friend bool operator==(const self &lhs, const self &rhs) {
    return lhs.v_ == rhs.v_;
  }
  constexpr friend bool operator!=(const self &lhs, const self &rhs) {
    return lhs.v_ != rhs.v_;
  }
};
}  // namespace MODINT
using mint = MODINT::Mint<998244353>;
vector<mint> fact, inv_fact;
void init_fact(int64_t n = N) {
  fact.resize(n + 1);
  inv_fact.resize(n + 1);
  fact[0] = fact[1] = inv_fact[0] = inv_fact[1] = 1;
  for_(i, 2, n) fact[i] = fact[i - 1] * i;
  inv_fact.back() = 1 / fact.back();
  rfor_(i, n - 1, 2) inv_fact[i] = inv_fact[i + 1] * (i + 1);
}
constexpr mint qpow(mint a, i64 b) {
  mint res(1);
  for (; b; b >>= 1, a *= a)
    if (b & 1) res *= a;
  return res;
}
auto inv_linear_gen = [](size_t n) -> std::vector<mint> {
  vector<mint> inv(n + 1, 1);
  for (int i = 2; i <= n; ++i)
    inv[i] = (mint::mod() - mint::mod() / i) * inv[mint::mod() % i];
  return inv;
};
vector<mint> inv = inv_linear_gen(N);
constexpr mint mPn(int m, int n) {
  return m < n || n < 0 ? 0 : fact[m] * inv_fact[m - n];
}
constexpr mint mCn(int m, int n) {
  return m < n || n < 0 ? 0 : mPn(m, n) * inv_fact[n];
}
vector<mint> pows;
void init_pows(int64_t m, int64_t n) {
  pows.resize(m + 1);
  for_(i, 0, m) pows[i] = qpow(i, n);
}
namespace Polynomial {
using data_t = int32_t;
using ldata_t = int64_t;
const size_t N = 1 << 20 | 500;
const data_t MOD = 998244353;
using udata_t = std::make_unsigned<data_t>::type;
using ludata_t = std::make_unsigned<ldata_t>::type;
const size_t DEG_LIMIT = N << 1;
namespace Helper {
constexpr ldata_t qpow(ldata_t a, ldata_t b, const ldata_t &mod) {
  ldata_t res(1);
  for (; b; b >>= 1, (a *= a) %= mod)
    if (b & 1) (res *= a) %= mod;
  return res;
}
constexpr ldata_t inverse(ldata_t n, const ldata_t &mod) {
  ldata_t b = mod, m0 = 0;
  for (ldata_t q = 0, _ = 0, m1 = 1; n;) {
    _ = b - n * (q = b / n);
    b = n;
    n = _;
    _ = m0 - m1 * q;
    m0 = m1;
    m1 = _;
  }
  return (m0 + (m0 < 0 ? mod / b : 0)) % mod;
}
constexpr data_t proot_impl_(data_t m) {
  if (m == 2) return 1;
  if (m == 3 || m == 5) return 2;
  if (m == 104857601 || m == 167772161 || m == 469762049) return 3;
  if (m == 754974721) return 11;
  if (m == 998244353 || m == 1004535809) return 3;
  data_t divs[20] = {2};
  data_t cnt = 1, x = (m - 1) / 2;
  while (!(x & 1)) x >>= 1;
  for (data_t i = 3; (ldata_t)i * i <= x; i += 2)
    if (x % i == 0) {
      divs[++cnt] = i;
      while (x % i == 0) x /= i;
    }
  if (x > 1) divs[++cnt] = x;
  for (data_t g = 2;; ++g) {
    bool ok = true;
    for (data_t i = 0; i < cnt; ++i)
      if (qpow(g, (m - 1) / divs[i], m) == 1) {
        ok = false;
        break;
      }
    if (ok) return g;
  }
}
template <data_t M>
constexpr data_t proot = proot_impl_(M);
constexpr int legendre_symbol(uint64_t a, uint64_t p) noexcept {
  if (a == 0) return 0;
  int s = 1, _ctz = 0;
  while (a > 1) {
    if (a == p || a == 0 || p < 2) return 0;
    _ctz = __builtin_ctzll(a);
    if (((p - 1) & 7) && ((p + 1) & 7) && (_ctz & 1)) s = -s;
    if ((a >>= _ctz) == 1) break;
    if ((((p - 1) & 7) * (a - 1)) & 7) s = -s;
    p %= a;
    auto _ = a;
    a = p;
    p = _;
  }
  return s;
}
struct GaussInt {
  data_t real, imag;
  const data_t i_sqr, mod;
  constexpr GaussInt &operator*=(GaussInt rhs) {
    const ldata_t _r = real, _i = imag;
    real =
      (data_t)((_r * rhs.real % mod + i_sqr * _i % mod * rhs.imag % mod) % mod);
    imag = (data_t)((_i * rhs.real % mod + _r * rhs.imag % mod) % mod);
    return *this;
  }
};
std::mt19937 eng__(time(nullptr));
data_t quad_residue(data_t n, data_t p) {
  if (n == 0 || n == 1 || n == p - 1) return n;
  if (legendre_symbol(n, p) != 1) return -1;
  std::uniform_int_distribution<ldata_t> u(2, p - 1);
  ldata_t a = u(eng__);
  while (legendre_symbol((a * a % p + p - n) % p, p) == 1) a = u(eng__);
  data_t ret = [](GaussInt a, udata_t b) {
    GaussInt res{1, 0, a.i_sqr, a.mod};
    for (; b; b >>= 1, a *= a)
      if (b & 1) res *= a;
    return res.real;
  }(GaussInt{(data_t)a, 1, (data_t)(a * a % p + p - n) % p, p}, (p + 1) / 2);
  return std::min(ret, p - ret);
}
template <size_t DEG_LIMIT, data_t MOD>
class INV_ {
protected:
  data_t data[DEG_LIMIT];

public:
  constexpr INV_() {
    data[0] = 0;
    data[1] = 1;
    for (size_t i = 2; i < DEG_LIMIT; ++i)
      data[i] = (data_t)((ldata_t)data[MOD % i] * (MOD - MOD / i) % MOD);
  }
  constexpr const data_t &operator[](size_t idx) const { return data[idx]; }
};
template <size_t DEG_LIMIT, data_t MOD>
class NTT_ {
  static constexpr data_t G = proot<MOD>, IG = inverse(G, MOD);

protected:
  data_t root[DEG_LIMIT];
  size_t rsz_;
  ludata_t f[DEG_LIMIT], w[DEG_LIMIT];
  constexpr void root_init(size_t n) {
    if (rsz_ == n) return;
    rsz_ = n;
    for (size_t i = 0; i < n; ++i)
      root[i] = (root[i >> 1] >> 1) | (data_t)((i & 1) * (n >> 1));
  }

public:
  constexpr NTT_() = default;
  constexpr void operator()(data_t *g, size_t n, bool inv = false) {
    root_init(n);
    w[0] = 1;
    for (size_t i = 0; i < n; ++i)
      f[i] = (((ldata_t)MOD << 5) + g[root[i]]) % MOD;
    for (size_t l = 1; l < n; l <<= 1) {
      ludata_t tG = qpow(inv ? IG : G, (MOD - 1) / (l + l), MOD);
      for (size_t i = 1; i < l; ++i) w[i] = w[i - 1] * tG % MOD;
      for (size_t k = 0; k < n; k += l + l)
        for (size_t p = 0; p < l; ++p) {
          ldata_t _ = w[p] * f[k | l | p] % MOD;
          f[k | l | p] = f[k | p] + (MOD - _);
          f[k | p] += _;
        }
      if (l == (1 << 10))
        for (size_t i = 0; i < n; ++i) f[i] %= MOD;
    }
    if (inv) {
      ludata_t in = inverse(n, MOD);
      for (size_t i = 0; i < n; ++i) g[i] = (data_t)(f[i] % MOD * in % MOD);
    } else
      for (size_t i = 0; i < n; ++i) g[i] = (data_t)(f[i] % MOD);
  }
};
const INV_<DEG_LIMIT, MOD> inv;
NTT_<DEG_LIMIT, MOD> NTT;
}  // namespace Helper
using Helper::inverse;
using Helper::NTT;
using Helper::qpow;
class Poly {
protected:
  std::vector<data_t> data;
  template <class Fodd, class Feven>
  void expand_base__(
    Poly &ans, size_t n, data_t val1, Fodd &&fodd, Feven &&feven) const {
    if (n == 1) {
      ans.data.push_back(val1);
      return;
    }
    if (n & 1) {
      expand_base__(ans, n - 1, val1, fodd, feven);
      fodd(ans, n);
      return;
    }
    expand_base__(ans, n / 2, val1, fodd, feven);
    feven(ans, n);
  }
  void inv_(Poly &ans, size_t n) const {
    expand_base__(
      ans,
      n,
      (data_t)inverse(data[0], MOD),
      [this](Poly &ans, size_t n) -> void {
        --n;
        ldata_t _ = 0;
        for (size_t i = 0; i < n; ++i)
          _ = (_ + (ldata_t)ans[i] * data[n - i]) % MOD;
        ans.data.push_back((data_t)(_ * inverse(MOD - data[0], MOD) % MOD));
      },
      [this](Poly &ans, size_t n) -> void {
        Poly sA = *this;
        sA.resize(n);
        ans = ans * 2 - (ans * ans * sA).resize(n);
      });
  }
  void exp_(Poly &ans, size_t n) const {
    expand_base__(
      ans,
      n,
      1,
      [this](Poly &ans, size_t n) -> void {
        n -= 2;
        ldata_t _ = 0;
        for (size_t i = 0; i <= n; ++i)
          _ = (_ + (i + 1) * data[i + 1] % MOD * ans[n - i] % MOD) % MOD;
        ans.data.push_back((data_t)(_ * Helper::inv[n + 1] % MOD));
      },
      [this](Poly &ans, size_t n) -> void {
        Poly ans_log = ans;
        ans_log.resize(n);
        ans_log.do_log();
        for (size_t i = 0; i < ans_log.size(); ++i)
          ans_log[i] = (MOD + data[i] - ans_log[i]) % MOD;
        ++ans_log[0];
        (ans *= ans_log).resize(n);
      });
  }

public:
  explicit Poly(decltype(DEG_LIMIT) sz = 0): data(std::min(DEG_LIMIT, sz)) {}
  explicit Poly(const std::initializer_list<data_t> &v): data(v) {}
  explicit Poly(const std::vector<data_t> &v): data(v) {}
  friend std::istream &operator>>(std::istream &is, Poly &poly) {
    for (auto &val : poly.data) is >> val;
    return is;
  }
  friend std::ostream &operator<<(std::ostream &os, const Poly &poly) {
    for (size_t i = 1; i < poly.size(); ++i) os << poly[i - 1] << ' ';
    return os << poly.data.back();
  }
  data_t &operator[](size_t x) { return data[x]; }
  const data_t &operator[](size_t x) const { return data[x]; }
  size_t size() const { return data.size(); }
  Poly &resize(size_t size) {
    data.resize(size);
    return *this;
  }
  Poly &operator*=(const data_t &c) {
    for (data_t &val : data) val = (data_t)((ldata_t)val * c % MOD);
    return *this;
  }
  friend Poly operator*(Poly poly, const data_t &c) { return poly *= c; }
  friend Poly operator*(const data_t &c, Poly poly) { return poly *= c; }
#define OOCR_(op, ...)                                                       \
  Poly &operator op##=(const Poly &rhs) __VA_ARGS__ friend Poly operator op( \
    Poly lhs, const Poly &rhs) {                                             \
    return lhs op## = rhs;                                                   \
  }
#define OO_(op, ...)                                                  \
  Poly &operator op##=(Poly rhs) __VA_ARGS__ friend Poly operator op( \
    Poly lhs, const Poly &rhs) {                                      \
    return lhs op## = rhs;                                            \
  }
  OOCR_(+, {
    resize(std::max(size(), rhs.size()));
    for (size_t i = 0; i < rhs.size(); ++i) {
      data[i] += rhs[i];
      data[i] -= data[i] >= MOD ? MOD : 0;
    }
    return *this;
  })
  OOCR_(-, {
    resize(std::max(size(), rhs.size()));
    for (size_t i = 0; i < rhs.size(); ++i) {
      data[i] += MOD - rhs[i];
      data[i] -= data[i] >= MOD ? MOD : 0;
    }
    return *this;
  })
  OOCR_(*, {
    static data_t a__[N << 1], b__[N << 1];
    std::copy(data.begin(), data.end(), a__);
    std::copy(rhs.data.begin(), rhs.data.end(), b__);
    size_t _sz = size();
    data.clear();
    resize(_sz + rhs.size() - 1);
    size_t n =
      (size_t)(1) << (size_t)std::max(1., std::ceil(std::log2(size())));
    NTT(a__, n);
    NTT(b__, n);
    for (size_t i = 0; i < n; ++i)
      a__[i] = (data_t)((ldata_t)a__[i] * b__[i] % MOD);
    NTT(a__, n, true);
    std::copy(a__, a__ + size(), data.begin());
    memset(a__, 0, sizeof(a__[0]) * (n));
    memset(b__, 0, sizeof(b__[0]) * (n));
    return *this;
  })
  OO_(/, {
    size_t n_ = size(), m_ = rhs.size();
    std::reverse(data.begin(), data.end());
    std::reverse(rhs.data.begin(), rhs.data.end());
    rhs.resize(n_ - m_ + 1);
    *this *= rhs.do_inverse();
    resize(n_ - m_ + 1);
    std::reverse(data.begin(), data.end());
    return *this;
  })
#undef OO_
#undef OOCR_
#define FUNC_(name, ...)                                      \
  Poly &do_##name() __VA_ARGS__ friend Poly name(Poly poly) { \
    return poly.do_##name();                                  \
  }
#define FUNCP2_(name, type1, var1, type2, var2, ...)                    \
  Poly &do_##name(type1 var1, type2 var2) __VA_ARGS__ friend Poly name( \
    Poly poly, type1 var1, type2 var2) {                                \
    return poly.do_##name(var1, var2);                                  \
  }
  FUNC_(inverse, {
    Poly ret;
    inv_(ret, size());
    return *this = ret;
  })
  FUNC_(derivative, {
    for (size_t i = 1; i < size(); ++i)
      data[i - 1] = (data_t)((ldata_t)data[i] * i % MOD);
    data.pop_back();
    return *this;
  })
  FUNC_(integral, {
    data.push_back(0);
    for (size_t i = size() - 1; i; --i)
      data[i] = (data_t)((ldata_t)data[i - 1] * Helper::inv[i] % MOD);
    data.front() = 0;
    return *this;
  })
  FUNC_(log, {
    size_t sz_ = size();
    *this = (derivative(*this) * inverse(*this)).do_integral();
    resize(sz_);
    return *this;
  })
  FUNC_(exp, {
    Poly ret;
    exp_(ret, size());
    return *this = ret;
  })
#undef FUNC_
#undef FUNCP2_
};
}  // namespace Polynomial
using Polynomial::Poly;
Poly dgf_stirling2;
void init_stirling2(int64_t m, int64_t n) {
  Poly a(m + 1), b(m + 1);
  for_(i, 0, m) a[i] = (pows[i] * inv_fact[i]).val();
  for_(i, 0, m) b[i] = (((i & 1) ? -1 : 1) * inv_fact[i]).val();
  (dgf_stirling2 = a * b).resize(m + 1);
}
}  // namespace Common
namespace solver1 {
using namespace Common;
mint main(i64 n, i64 m) { return qpow(m, n); }
}  // namespace solver1
namespace solver2 {
using namespace Common;
mint main(i64 n, i64 m) { return mPn(m, n); }
}  // namespace solver2
namespace solver3 {
using namespace Common;
mint main(i64 n, i64 m) {
  mint ans = 0;
  int f = m & 1;
  for_(i, 1, m) {
    f ? (ans += mCn(m, i) * pows[i]) : (ans -= mCn(m, i) * pows[i]);
    f ^= 1;
  }
  return ans;
}
}  // namespace solver3
namespace solver4 {
using namespace Common;
mint main(i64 n, i64 m) {
  mint ans = 0;
  for_(i, 0, m) ans += dgf_stirling2[i];
  return ans;
}
}  // namespace solver4
namespace solver5 {
using namespace Common;
mint main(i64 n, i64 m) { return m >= n; }
}  // namespace solver5
namespace solver6 {
using namespace Common;
mint main(i64 n, i64 m) { return dgf_stirling2[m]; }
}  // namespace solver6
namespace solver7 {
using namespace Common;
mint main(i64 n, i64 m) { return mCn(m + n - 1, m - 1); }
}  // namespace solver7
namespace solver8 {
using namespace Common;
mint main(i64 n, i64 m) { return mCn(m, n); }
}  // namespace solver8
namespace solver9 {
using namespace Common;
mint main(i64 n, i64 m) { return mCn(n - 1, m - 1); }
}  // namespace solver9
namespace solver10 {
using namespace Common;
Poly f;
void init_f(int64_t m, int64_t n) {
  f.resize(n + 1);
  for_(i, 1, m)
    for_(k, 1, n / i) f[i * k] = (f[i * k] + inv[k]).val();
  f.do_exp();
}
mint main(i64 n, i64 m) { return f[n]; }
}  // namespace solver10
namespace solver11 {
using namespace Common;
mint main(i64 n, i64 m) { return m >= n; }
}  // namespace solver11
namespace solver12 {
using namespace Common;
mint main(i64 n, i64 m) { return n >= m ? solver10::f[n - m] : 0; }
}  // namespace solver12
using Func = Common::mint (*)(i64, i64);
const Func solver[12] = {solver1::main,
                         solver2::main,
                         solver3::main,
                         solver4::main,
                         solver5::main,
                         solver6::main,
                         solver7::main,
                         solver8::main,
                         solver9::main,
                         solver10::main,
                         solver11::main,
                         solver12::main};
auto solve([[maybe_unused]] int t_ = 0) -> void {
  i64 n, m;
  cin >> n >> m;
  Common::init_fact();
  Common::init_pows(m, n);
  Common::init_stirling2(m, n);
  solver10::init_f(m, n);
  for (auto &&f : solver) cout << f(n, m) << '\n';
}
int main() {
  std::ios::sync_with_stdio(false);
  std::cin.tie(nullptr);
  int i_ = 0;
  solve(i_);
  return 0;
}