cp-library-cpp
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test/src/polynomial/rook_polynomial/dummy.test.cpp
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- Last update: 2024-01-31 02:46:42+09:00
- Problem: https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A
Depends on
- Cartesian Tree
(library/datastructure/cartesian_tree.hpp)
- 階乗テーブル
(library/math/factorial.hpp)
- 逆元テーブル
(library/math/inv_mods.hpp)
- Modint Extension
(library/math/modint_extension.hpp)
- Convert To Newton Basis
(library/polynomial/convert_to_newton_basis.hpp)
- Formal Power Series
(library/polynomial/formal_power_series.hpp)
- FFT-free な形式的べき級数
(library/polynomial/fps_naive.hpp)
- library/polynomial/rook_polynomial.hpp
- Type Traits
(library/type_traits/type_traits.hpp)
Code
#define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A"
#include <iostream>
#include "library/polynomial/rook_polynomial.hpp"
#include "library/polynomial/formal_power_series.hpp"
#include <atcoder/modint>
using mint = atcoder::modint998244353;
namespace atcoder {
std::istream& operator>>(std::istream& in, mint &a) {
long long e; in >> e; a = e;
return in;
}
std::ostream& operator<<(std::ostream& out, const mint &a) {
out << a.val();
return out;
}
} // namespace atcoder
using fps = suisen::FormalPowerSeries<mint>;
#include <map>
fps naive(const std::vector<int> &h) {
std::map<int, fps> pd;
pd[0] = { 1 };
for (int k : h) {
std::map<int, fps> dp;
for (auto [mask, f] : pd) {
const int nxt_mask = mask ^ ((mask >> k) << k);
dp[nxt_mask] += f;
for (int pos = 0; pos < k; ++pos) if (not ((nxt_mask >> pos) & 1)) {
dp[nxt_mask | (1 << pos)] += f << 1;
}
}
pd.swap(dp);
}
fps res(h.size() + 1);
for (auto [_, f] : pd) res += f;
return res;
}
#include <cassert>
void test() {
const std::vector<int> h { 6, 2, 5, 3, 2, 4, 5, 2, 3 };
const int n = h.size();
fps f = suisen::rook_polynomial_skyline_board<fps>(h);
fps g = naive(h);
assert(f == g);
}
int main() {
test();
std::cout << "Hello World" << std::endl;
return 0;
}#line 1 "test/src/polynomial/rook_polynomial/dummy.test.cpp"
#define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A"
#include <iostream>
#line 1 "library/polynomial/rook_polynomial.hpp"
#include <algorithm>
#include <cassert>
#include <numeric>
#include <vector>
#line 1 "library/polynomial/convert_to_newton_basis.hpp"
#include <tuple>
#line 6 "library/polynomial/convert_to_newton_basis.hpp"
namespace suisen {
// Returns b=(b_0,...,b_{N-1}) s.t. f(x) = Sum[i=0,N-1] b_i Prod[j=0,i-1](x - p_j)
template <typename FPSType>
std::vector<typename FPSType::value_type> convert_to_newton_basis(const FPSType& f, const std::vector<typename FPSType::value_type>& p) {
const int n = p.size();
assert(f.size() == n);
int m = 1;
while (m < n) m <<= 1;
std::vector<FPSType> seg(2 * m);
for (int i = 0; i < m; ++i) {
seg[m + i] = { i < n ? -p[i] : 0, 1 };
}
for (int i = m - 1; i > 0; --i) {
if (((i + 1) & -(i + 1)) == (i + 1)) continue; // i = 2^k - 1
seg[i] = seg[2 * i] * seg[2 * i + 1];
}
seg[1] = f;
for (int i = 1; i < m; ++i) {
std::tie(seg[2 * i + 1], seg[2 * i]) = seg[i].div_mod(seg[2 * i]);
}
std::vector<typename FPSType::value_type> b(n);
for (int i = 0; i < n; ++i) {
b[i] = seg[m + i].safe_get(0);
}
return b;
}
} // namespace suisen
#line 1 "library/datastructure/cartesian_tree.hpp"
#include <array>
#line 6 "library/datastructure/cartesian_tree.hpp"
#include <functional>
#line 8 "library/datastructure/cartesian_tree.hpp"
namespace suisen {
struct CartesianTree : public std::vector<std::array<int, 2>> {
using base_type = std::vector<std::array<int, 2>>;
static constexpr int absent = -1;
const int root;
CartesianTree() : base_type(), root(0) {}
CartesianTree(int root, const base_type& g) : base_type(g), root(root) {}
CartesianTree(int root, base_type&& g) : base_type(std::move(g)), root(root) {}
auto ranges() const {
std::vector<std::pair<int, int>> res;
res.reserve(size());
auto rec = [&](auto rec, int l, int m, int r) -> void {
if (m == absent) return;
const auto& [lm, rm] = (*this)[m];
rec(rec, l, lm, m), res.emplace_back(l, r), rec(rec, m + 1, rm, r);
};
rec(rec, 0, root, size());
return res;
}
};
template <typename Comparator>
struct CartesianTreeBuilder {
CartesianTreeBuilder() = default;
template <typename RandomAccessibleContainer>
CartesianTreeBuilder(const RandomAccessibleContainer& a, Comparator comp = Comparator{}) : n(a.size()), comp(comp), par(calc_par(a, comp)) {}
CartesianTree build() const {
int root = -1;
std::vector<std::array<int, 2>> g(n, { CartesianTree::absent, CartesianTree::absent });
for (int i = 0; i < n; ++i) {
int p = par[i];
(p >= 0 ? g[p][p <= i] : root) = i;
}
return CartesianTree{ root, std::move(g) };
}
template <typename RandomAccessibleContainer>
static CartesianTree build(const RandomAccessibleContainer& a, Comparator comp = Comparator{}) {
return CartesianTreeBuilder(a, comp).build();
}
int parent(std::size_t i) const {
assert(i < std::size_t(n));
return par[i];
}
int operator[](std::size_t i) const {
return parent(i);
}
private:
const int n;
const Comparator comp;
const std::vector<int> par;
template <typename RandomAccessibleContainer>
static std::vector<int> calc_par(const RandomAccessibleContainer& a, Comparator comp) {
const int n = a.size();
std::vector<int> par(n, -1);
for (int i = 1; i < n; ++i) {
int p = i - 1, l = i;
while (p >= 0 and comp(a[i], a[p])) l = p, p = par[p];
par[l] = i, par[i] = p;
}
return par;
}
};
using MinCartesianTreeBuilder = CartesianTreeBuilder<std::less<>>;
using MaxCartesianTreeBuilder = CartesianTreeBuilder<std::greater<>>;
} // namespace suisen
#line 1 "library/math/factorial.hpp"
#line 6 "library/math/factorial.hpp"
namespace suisen {
template <typename T, typename U = T>
struct factorial {
factorial() = default;
factorial(int n) { ensure(n); }
static void ensure(const int n) {
int sz = _fac.size();
if (n + 1 <= sz) return;
int new_size = std::max(n + 1, sz * 2);
_fac.resize(new_size), _fac_inv.resize(new_size);
for (int i = sz; i < new_size; ++i) _fac[i] = _fac[i - 1] * i;
_fac_inv[new_size - 1] = U(1) / _fac[new_size - 1];
for (int i = new_size - 1; i > sz; --i) _fac_inv[i - 1] = _fac_inv[i] * i;
}
T fac(const int i) {
ensure(i);
return _fac[i];
}
T operator()(int i) {
return fac(i);
}
U fac_inv(const int i) {
ensure(i);
return _fac_inv[i];
}
U binom(const int n, const int r) {
if (n < 0 or r < 0 or n < r) return 0;
ensure(n);
return _fac[n] * _fac_inv[r] * _fac_inv[n - r];
}
template <typename ...Ds, std::enable_if_t<std::conjunction_v<std::is_integral<Ds>...>, std::nullptr_t> = nullptr>
U polynom(const int n, const Ds& ...ds) {
if (n < 0) return 0;
ensure(n);
int sumd = 0;
U res = _fac[n];
for (int d : { ds... }) {
if (d < 0 or d > n) return 0;
sumd += d;
res *= _fac_inv[d];
}
if (sumd > n) return 0;
res *= _fac_inv[n - sumd];
return res;
}
U perm(const int n, const int r) {
if (n < 0 or r < 0 or n < r) return 0;
ensure(n);
return _fac[n] * _fac_inv[n - r];
}
private:
static std::vector<T> _fac;
static std::vector<U> _fac_inv;
};
template <typename T, typename U>
std::vector<T> factorial<T, U>::_fac{ 1 };
template <typename T, typename U>
std::vector<U> factorial<T, U>::_fac_inv{ 1 };
} // namespace suisen
#line 12 "library/polynomial/rook_polynomial.hpp"
namespace suisen {
// O(n(log n)^2). returns rook polynomial r s.t. r[k] := # ways to put k non-attacking rooks on a ferrers board
template <typename FPSType>
FPSType rook_polynomial_ferrers_board(const std::vector<int> &h) {
using mint = typename FPSType::value_type;
assert(std::is_sorted(h.begin(), h.end()));
const int n = h.size();
std::vector<FPSType> fs(n);
for (int i = 0; i < n; ++i) fs[i] = FPSType{ h[i] - i, 1 };
FPSType f = FPSType::prod(fs);
std::vector<mint> p(n + 1);
std::iota(p.begin(), p.end(), 0);
FPSType r = convert_to_newton_basis(f, p);
std::reverse(r.begin(), r.end());
return r;
}
// O(n^2 log n). returns rook polynomial r s.t. r[k] := # ways to put k non-attacking rooks on a skyline board
template <typename FPSType>
FPSType rook_polynomial_skyline_board(const std::vector<int> &h) {
using fps = FPSType;
using mint = typename fps::value_type;
const int n = h.size();
factorial<mint> fac(n);
MinCartesianTreeBuilder ct_builder{h};
CartesianTree t = ct_builder.build();
auto dfs = [&](auto dfs, int u, int l, int r) -> fps {
if (u == t.absent) return { 1 };
fps f = dfs(dfs, t[u][0], l, u);
fps g = dfs(dfs, t[u][1], u + 1, r);
fps fg = f * g; // O(n^2)
fg.push_back(0);
const int a = h[u] - (u == t.root ? 0 : h[ct_builder.parent(u)]);
const int b = r - l;
assert(int(fg.size()) == b + 1);
fps s(b + 1), t(b + 1);
mint binom_a_i = 1;
for (int i = 0; i <= b; ++i) {
s[i] = fg[i] * fac.fac(b - i);
t[i] = binom_a_i;
binom_a_i *= (a - i) * fac.fac_inv(i + 1) * fac.fac(i);
}
fps st = s * t;
st.resize(b + 1);
for (int i = 0; i <= b; ++i) {
st[i] *= fac.fac_inv(b - i);
}
return st;
};
return dfs(dfs, t.root, 0, n);
}
} // namespace suisen
#line 1 "library/polynomial/formal_power_series.hpp"
#include <limits>
#include <optional>
#include <queue>
#include <atcoder/modint>
#include <atcoder/convolution>
#line 1 "library/polynomial/fps_naive.hpp"
#line 5 "library/polynomial/fps_naive.hpp"
#include <cmath>
#line 7 "library/polynomial/fps_naive.hpp"
#include <type_traits>
#line 9 "library/polynomial/fps_naive.hpp"
#line 1 "library/type_traits/type_traits.hpp"
#line 7 "library/type_traits/type_traits.hpp"
namespace suisen {
template <typename ...Constraints> using constraints_t = std::enable_if_t<std::conjunction_v<Constraints...>, std::nullptr_t>;
template <typename T, typename = std::nullptr_t> struct bitnum { static constexpr int value = 0; };
template <typename T> struct bitnum<T, constraints_t<std::is_integral<T>>> { static constexpr int value = std::numeric_limits<std::make_unsigned_t<T>>::digits; };
template <typename T> static constexpr int bitnum_v = bitnum<T>::value;
template <typename T, size_t n> struct is_nbit { static constexpr bool value = bitnum_v<T> == n; };
template <typename T, size_t n> static constexpr bool is_nbit_v = is_nbit<T, n>::value;
template <typename T, typename = std::nullptr_t> struct safely_multipliable { using type = T; };
template <typename T> struct safely_multipliable<T, constraints_t<std::is_signed<T>, is_nbit<T, 32>>> { using type = long long; };
template <typename T> struct safely_multipliable<T, constraints_t<std::is_signed<T>, is_nbit<T, 64>>> { using type = __int128_t; };
template <typename T> struct safely_multipliable<T, constraints_t<std::is_unsigned<T>, is_nbit<T, 32>>> { using type = unsigned long long; };
template <typename T> struct safely_multipliable<T, constraints_t<std::is_unsigned<T>, is_nbit<T, 64>>> { using type = __uint128_t; };
template <typename T> using safely_multipliable_t = typename safely_multipliable<T>::type;
template <typename T, typename = void> struct rec_value_type { using type = T; };
template <typename T> struct rec_value_type<T, std::void_t<typename T::value_type>> {
using type = typename rec_value_type<typename T::value_type>::type;
};
template <typename T> using rec_value_type_t = typename rec_value_type<T>::type;
template <typename T> class is_iterable {
template <typename T_> static auto test(T_ e) -> decltype(e.begin(), e.end(), std::true_type{});
static std::false_type test(...);
public:
static constexpr bool value = decltype(test(std::declval<T>()))::value;
};
template <typename T> static constexpr bool is_iterable_v = is_iterable<T>::value;
template <typename T> class is_writable {
template <typename T_> static auto test(T_ e) -> decltype(std::declval<std::ostream&>() << e, std::true_type{});
static std::false_type test(...);
public:
static constexpr bool value = decltype(test(std::declval<T>()))::value;
};
template <typename T> static constexpr bool is_writable_v = is_writable<T>::value;
template <typename T> class is_readable {
template <typename T_> static auto test(T_ e) -> decltype(std::declval<std::istream&>() >> e, std::true_type{});
static std::false_type test(...);
public:
static constexpr bool value = decltype(test(std::declval<T>()))::value;
};
template <typename T> static constexpr bool is_readable_v = is_readable<T>::value;
} // namespace suisen
#line 11 "library/polynomial/fps_naive.hpp"
#line 1 "library/math/modint_extension.hpp"
#line 6 "library/math/modint_extension.hpp"
/**
* refernce: https://37zigen.com/tonelli-shanks-algorithm/
* calculates x s.t. x^2 = a mod p in O((log p)^2).
*/
template <typename mint>
std::optional<mint> safe_sqrt(mint a) {
static int p = mint::mod();
if (a == 0) return std::make_optional(0);
if (p == 2) return std::make_optional(a);
if (a.pow((p - 1) / 2) != 1) return std::nullopt;
mint b = 1;
while (b.pow((p - 1) / 2) == 1) ++b;
static int tlz = __builtin_ctz(p - 1), q = (p - 1) >> tlz;
mint x = a.pow((q + 1) / 2);
b = b.pow(q);
for (int shift = 2; x * x != a; ++shift) {
mint e = a.inv() * x * x;
if (e.pow(1 << (tlz - shift)) != 1) x *= b;
b *= b;
}
return std::make_optional(x);
}
/**
* calculates x s.t. x^2 = a mod p in O((log p)^2).
* if not exists, raises runtime error.
*/
template <typename mint>
auto sqrt(mint a) -> decltype(mint::mod(), mint()) {
return *safe_sqrt(a);
}
template <typename mint>
auto log(mint a) -> decltype(mint::mod(), mint()) {
assert(a == 1);
return 0;
}
template <typename mint>
auto exp(mint a) -> decltype(mint::mod(), mint()) {
assert(a == 0);
return 1;
}
template <typename mint, typename T>
auto pow(mint a, T b) -> decltype(mint::mod(), mint()) {
return a.pow(b);
}
template <typename mint>
auto inv(mint a) -> decltype(mint::mod(), mint()) {
return a.inv();
}
#line 1 "library/math/inv_mods.hpp"
#line 5 "library/math/inv_mods.hpp"
namespace suisen {
template <typename mint>
class inv_mods {
public:
inv_mods() = default;
inv_mods(int n) { ensure(n); }
const mint& operator[](int i) const {
ensure(i);
return invs[i];
}
static void ensure(int n) {
int sz = invs.size();
if (sz < 2) invs = { 0, 1 }, sz = 2;
if (sz < n + 1) {
invs.resize(n + 1);
for (int i = sz; i <= n; ++i) invs[i] = mint(mod - mod / i) * invs[mod % i];
}
}
private:
static std::vector<mint> invs;
static constexpr int mod = mint::mod();
};
template <typename mint>
std::vector<mint> inv_mods<mint>::invs{};
template <typename mint>
std::vector<mint> get_invs(const std::vector<mint>& vs) {
const int n = vs.size();
mint p = 1;
for (auto& e : vs) {
p *= e;
assert(e != 0);
}
mint ip = p.inv();
std::vector<mint> rp(n + 1);
rp[n] = 1;
for (int i = n - 1; i >= 0; --i) {
rp[i] = rp[i + 1] * vs[i];
}
std::vector<mint> res(n);
for (int i = 0; i < n; ++i) {
res[i] = ip * rp[i + 1];
ip *= vs[i];
}
return res;
}
}
#line 14 "library/polynomial/fps_naive.hpp"
namespace suisen {
template <typename T>
struct FPSNaive : std::vector<T> {
static inline int MAX_SIZE = std::numeric_limits<int>::max() / 2;
using value_type = T;
using element_type = rec_value_type_t<T>;
using std::vector<value_type>::vector;
FPSNaive(const std::initializer_list<value_type> l) : std::vector<value_type>::vector(l) {}
FPSNaive(const std::vector<value_type>& v) : std::vector<value_type>::vector(v) {}
static void set_max_size(int n) {
FPSNaive<T>::MAX_SIZE = n;
}
const value_type operator[](int n) const {
return n <= deg() ? unsafe_get(n) : value_type{ 0 };
}
value_type& operator[](int n) {
return ensure_deg(n), unsafe_get(n);
}
int size() const {
return std::vector<value_type>::size();
}
int deg() const {
return size() - 1;
}
int normalize() {
while (size() and this->back() == value_type{ 0 }) this->pop_back();
return deg();
}
FPSNaive& cut_inplace(int n) {
if (size() > n) this->resize(std::max(0, n));
return *this;
}
FPSNaive cut(int n) const {
FPSNaive f = FPSNaive(*this).cut_inplace(n);
return f;
}
FPSNaive operator+() const {
return FPSNaive(*this);
}
FPSNaive operator-() const {
FPSNaive f(*this);
for (auto& e : f) e = -e;
return f;
}
FPSNaive& operator++() { return ++(*this)[0], * this; }
FPSNaive& operator--() { return --(*this)[0], * this; }
FPSNaive& operator+=(const value_type x) { return (*this)[0] += x, *this; }
FPSNaive& operator-=(const value_type x) { return (*this)[0] -= x, *this; }
FPSNaive& operator+=(const FPSNaive& g) {
ensure_deg(g.deg());
for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) += g.unsafe_get(i);
return *this;
}
FPSNaive& operator-=(const FPSNaive& g) {
ensure_deg(g.deg());
for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) -= g.unsafe_get(i);
return *this;
}
FPSNaive& operator*=(const FPSNaive& g) { return *this = *this * g; }
FPSNaive& operator*=(const value_type x) {
for (auto& e : *this) e *= x;
return *this;
}
FPSNaive& operator/=(const FPSNaive& g) { return *this = *this / g; }
FPSNaive& operator%=(const FPSNaive& g) { return *this = *this % g; }
FPSNaive& operator<<=(const int shamt) {
this->insert(this->begin(), shamt, value_type{ 0 });
return *this;
}
FPSNaive& operator>>=(const int shamt) {
if (shamt > size()) this->clear();
else this->erase(this->begin(), this->begin() + shamt);
return *this;
}
friend FPSNaive operator+(FPSNaive f, const FPSNaive& g) { f += g; return f; }
friend FPSNaive operator+(FPSNaive f, const value_type& x) { f += x; return f; }
friend FPSNaive operator-(FPSNaive f, const FPSNaive& g) { f -= g; return f; }
friend FPSNaive operator-(FPSNaive f, const value_type& x) { f -= x; return f; }
friend FPSNaive operator*(const FPSNaive& f, const FPSNaive& g) {
if (f.empty() or g.empty()) return FPSNaive{};
const int n = f.size(), m = g.size();
FPSNaive h(std::min(MAX_SIZE, n + m - 1));
for (int i = 0; i < n; ++i) for (int j = 0; j < m; ++j) {
if (i + j >= MAX_SIZE) break;
h.unsafe_get(i + j) += f.unsafe_get(i) * g.unsafe_get(j);
}
return h;
}
friend FPSNaive operator*(FPSNaive f, const value_type& x) { f *= x; return f; }
friend FPSNaive operator/(FPSNaive f, const FPSNaive& g) { return std::move(f.div_mod(g).first); }
friend FPSNaive operator%(FPSNaive f, const FPSNaive& g) { return std::move(f.div_mod(g).second); }
friend FPSNaive operator*(const value_type x, FPSNaive f) { f *= x; return f; }
friend FPSNaive operator<<(FPSNaive f, const int shamt) { f <<= shamt; return f; }
friend FPSNaive operator>>(FPSNaive f, const int shamt) { f >>= shamt; return f; }
std::pair<FPSNaive, FPSNaive> div_mod(FPSNaive g) const {
FPSNaive f = *this;
const int fd = f.normalize(), gd = g.normalize();
assert(gd >= 0);
if (fd < gd) return { FPSNaive{}, f };
if (gd == 0) return { f *= g.unsafe_get(0).inv(), FPSNaive{} };
const int k = f.deg() - gd;
value_type head_inv = g.unsafe_get(gd).inv();
FPSNaive q(k + 1);
for (int i = k; i >= 0; --i) {
value_type div = f.unsafe_get(i + gd) * head_inv;
q.unsafe_get(i) = div;
for (int j = 0; j <= gd; ++j) f.unsafe_get(i + j) -= div * g.unsafe_get(j);
}
f.cut_inplace(gd);
f.normalize();
return { q, f };
}
friend bool operator==(const FPSNaive& f, const FPSNaive& g) {
const int n = f.size(), m = g.size();
if (n < m) return g == f;
for (int i = 0; i < m; ++i) if (f.unsafe_get(i) != g.unsafe_get(i)) return false;
for (int i = m; i < n; ++i) if (f.unsafe_get(i) != 0) return false;
return true;
}
friend bool operator!=(const FPSNaive& f, const FPSNaive& g) {
return not (f == g);
}
FPSNaive mul(const FPSNaive& g, int n = -1) const {
if (n < 0) n = size();
if (this->empty() or g.empty()) return FPSNaive{};
const int m = size(), k = g.size();
FPSNaive h(std::min(n, m + k - 1));
for (int i = 0; i < m; ++i) {
for (int j = 0, jr = std::min(k, n - i); j < jr; ++j) {
h.unsafe_get(i + j) += unsafe_get(i) * g.unsafe_get(j);
}
}
return h;
}
FPSNaive diff() const {
if (this->empty()) return {};
FPSNaive g(size() - 1);
for (int i = 1; i <= deg(); ++i) g.unsafe_get(i - 1) = unsafe_get(i) * i;
return g;
}
FPSNaive intg() const {
const int n = size();
FPSNaive g(n + 1);
for (int i = 0; i < n; ++i) g.unsafe_get(i + 1) = unsafe_get(i) * invs[i + 1];
if (g.deg() > MAX_SIZE) g.cut_inplace(MAX_SIZE);
return g;
}
FPSNaive inv(int n = -1) const {
if (n < 0) n = size();
FPSNaive g(n);
const value_type inv_f0 = ::inv(unsafe_get(0));
g.unsafe_get(0) = inv_f0;
for (int i = 1; i < n; ++i) {
for (int j = 1; j <= i; ++j) g.unsafe_get(i) -= g.unsafe_get(i - j) * (*this)[j];
g.unsafe_get(i) *= inv_f0;
}
return g;
}
FPSNaive exp(int n = -1) const {
if (n < 0) n = size();
assert(unsafe_get(0) == value_type{ 0 });
FPSNaive g(n);
g.unsafe_get(0) = value_type{ 1 };
for (int i = 1; i < n; ++i) {
for (int j = 1; j <= i; ++j) g.unsafe_get(i) += j * g.unsafe_get(i - j) * (*this)[j];
g.unsafe_get(i) *= invs[i];
}
return g;
}
FPSNaive log(int n = -1) const {
if (n < 0) n = size();
assert(unsafe_get(0) == value_type{ 1 });
FPSNaive g(n);
g.unsafe_get(0) = value_type{ 0 };
for (int i = 1; i < n; ++i) {
g.unsafe_get(i) = i * (*this)[i];
for (int j = 1; j < i; ++j) g.unsafe_get(i) -= (i - j) * g.unsafe_get(i - j) * (*this)[j];
g.unsafe_get(i) *= invs[i];
}
return g;
}
FPSNaive pow(const long long k, int n = -1) const {
if (n < 0) n = size();
if (k == 0) {
FPSNaive res(n);
res[0] = 1;
return res;
}
int z = 0;
while (z < size() and unsafe_get(z) == value_type{ 0 }) ++z;
if (z == size() or z > (n - 1) / k) return FPSNaive(n, 0);
const int m = n - z * k;
FPSNaive g(m);
const value_type inv_f0 = ::inv(unsafe_get(z));
g.unsafe_get(0) = unsafe_get(z).pow(k);
for (int i = 1; i < m; ++i) {
for (int j = 1; j <= i; ++j) g.unsafe_get(i) += (element_type{ k } *j - (i - j)) * g.unsafe_get(i - j) * (*this)[z + j];
g.unsafe_get(i) *= inv_f0 * invs[i];
}
g <<= z * k;
return g;
}
std::optional<FPSNaive> safe_sqrt(int n = -1) const {
if (n < 0) n = size();
int dl = 0;
while (dl < size() and unsafe_get(dl) == value_type{ 0 }) ++dl;
if (dl == size()) return FPSNaive(n, 0);
if (dl & 1) return std::nullopt;
const int m = n - dl / 2;
FPSNaive g(m);
auto opt_g0 = ::safe_sqrt((*this)[dl]);
if (not opt_g0.has_value()) return std::nullopt;
g.unsafe_get(0) = *opt_g0;
value_type inv_2g0 = ::inv(2 * g.unsafe_get(0));
for (int i = 1; i < m; ++i) {
g.unsafe_get(i) = (*this)[dl + i];
for (int j = 1; j < i; ++j) g.unsafe_get(i) -= g.unsafe_get(j) * g.unsafe_get(i - j);
g.unsafe_get(i) *= inv_2g0;
}
g <<= dl / 2;
return g;
}
FPSNaive sqrt(int n = -1) const {
if (n < 0) n = size();
return *safe_sqrt(n);
}
value_type eval(value_type x) const {
value_type y = 0;
for (int i = size() - 1; i >= 0; --i) y = y * x + unsafe_get(i);
return y;
}
private:
static inline inv_mods<element_type> invs;
void ensure_deg(int d) {
if (deg() < d) this->resize(d + 1, value_type{ 0 });
}
const value_type& unsafe_get(int i) const {
return std::vector<value_type>::operator[](i);
}
value_type& unsafe_get(int i) {
return std::vector<value_type>::operator[](i);
}
};
} // namespace suisen
template <typename mint>
suisen::FPSNaive<mint> sqrt(suisen::FPSNaive<mint> a) {
return a.sqrt();
}
template <typename mint>
suisen::FPSNaive<mint> log(suisen::FPSNaive<mint> a) {
return a.log();
}
template <typename mint>
suisen::FPSNaive<mint> exp(suisen::FPSNaive<mint> a) {
return a.exp();
}
template <typename mint, typename T>
suisen::FPSNaive<mint> pow(suisen::FPSNaive<mint> a, T b) {
return a.pow(b);
}
template <typename mint>
suisen::FPSNaive<mint> inv(suisen::FPSNaive<mint> a) {
return a.inv();
}
#line 14 "library/polynomial/formal_power_series.hpp"
namespace suisen {
template <typename mint, atcoder::internal::is_static_modint_t<mint>* = nullptr>
struct FormalPowerSeries : std::vector<mint> {
using base_type = std::vector<mint>;
using value_type = typename base_type::value_type;
using base_type::vector;
FormalPowerSeries(const std::initializer_list<value_type> l) : std::vector<value_type>::vector(l) {}
FormalPowerSeries(const std::vector<value_type>& v) : std::vector<value_type>::vector(v) {}
int size() const noexcept {
return base_type::size();
}
int deg() const noexcept {
return size() - 1;
}
void ensure(int n) {
if (size() < n) this->resize(n);
}
value_type safe_get(int d) const {
return d <= deg() ? (*this)[d] : 0;
}
value_type& safe_get(int d) {
ensure(d + 1);
return (*this)[d];
}
FormalPowerSeries& cut_trailing_zeros() {
while (size() and this->back() == 0) this->pop_back();
return *this;
}
FormalPowerSeries& cut(int n) {
if (size() > n) this->resize(std::max(0, n));
return *this;
}
FormalPowerSeries cut_copy(int n) const {
FormalPowerSeries res(this->begin(), this->begin() + std::min(size(), n));
res.ensure(n);
return res;
}
FormalPowerSeries cut_copy(int l, int r) const {
if (l >= size()) return FormalPowerSeries(r - l, 0);
FormalPowerSeries res(this->begin() + l, this->begin() + std::min(size(), r));
res.ensure(r - l);
return res;
}
/* Unary Operations */
FormalPowerSeries operator+() const { return *this; }
FormalPowerSeries operator-() const {
FormalPowerSeries res = *this;
for (auto& e : res) e = -e;
return res;
}
FormalPowerSeries& operator++() { return ++safe_get(0), * this; }
FormalPowerSeries& operator--() { return --safe_get(0), * this; }
FormalPowerSeries operator++(int) {
FormalPowerSeries res = *this;
++(*this);
return res;
}
FormalPowerSeries operator--(int) {
FormalPowerSeries res = *this;
--(*this);
return res;
}
/* Binary Operations With Constant */
FormalPowerSeries& operator+=(const value_type& x) { return safe_get(0) += x, *this; }
FormalPowerSeries& operator-=(const value_type& x) { return safe_get(0) -= x, *this; }
FormalPowerSeries& operator*=(const value_type& x) {
for (auto& e : *this) e *= x;
return *this;
}
FormalPowerSeries& operator/=(const value_type& x) { return *this *= x.inv(); }
friend FormalPowerSeries operator+(FormalPowerSeries f, const value_type& x) { f += x; return f; }
friend FormalPowerSeries operator+(const value_type& x, FormalPowerSeries f) { f += x; return f; }
friend FormalPowerSeries operator-(FormalPowerSeries f, const value_type& x) { f -= x; return f; }
friend FormalPowerSeries operator-(const value_type& x, FormalPowerSeries f) { f -= x; return -f; }
friend FormalPowerSeries operator*(FormalPowerSeries f, const value_type& x) { f *= x; return f; }
friend FormalPowerSeries operator*(const value_type& x, FormalPowerSeries f) { f *= x; return f; }
friend FormalPowerSeries operator/(FormalPowerSeries f, const value_type& x) { f /= x; return f; }
/* Binary Operations With Formal Power Series */
FormalPowerSeries& operator+=(const FormalPowerSeries& g) {
const int n = g.size();
ensure(n);
for (int i = 0; i < n; ++i) (*this)[i] += g[i];
return *this;
}
FormalPowerSeries& operator-=(const FormalPowerSeries& g) {
const int n = g.size();
ensure(n);
for (int i = 0; i < n; ++i) (*this)[i] -= g[i];
return *this;
}
FormalPowerSeries& operator*=(const FormalPowerSeries& g) { return *this = *this * g; }
FormalPowerSeries& operator/=(const FormalPowerSeries& g) { return *this = *this / g; }
FormalPowerSeries& operator%=(const FormalPowerSeries& g) { return *this = *this % g; }
friend FormalPowerSeries operator+(FormalPowerSeries f, const FormalPowerSeries& g) { f += g; return f; }
friend FormalPowerSeries operator-(FormalPowerSeries f, const FormalPowerSeries& g) { f -= g; return f; }
friend FormalPowerSeries operator*(const FormalPowerSeries& f, const FormalPowerSeries& g) {
const int siz_f = f.size(), siz_g = g.size();
if (siz_f < siz_g) return g * f;
if (std::min(siz_f, siz_g) <= 60) return atcoder::convolution(f, g);
const int deg = siz_f + siz_g - 2;
int fpow2 = 1;
while ((fpow2 << 1) <= deg) fpow2 <<= 1;
if (const int dif = deg - fpow2 + 1; dif <= 10) {
FormalPowerSeries h = atcoder::convolution(std::vector<mint>(f.begin(), f.end() - dif), g);
h.resize(h.size() + dif);
for (int i = siz_f - dif; i < siz_f; ++i) for (int j = 0; j < siz_g; ++j) {
h[i + j] += f[i] * g[j];
}
return h;
}
return atcoder::convolution(f, g);
}
friend FormalPowerSeries operator/(FormalPowerSeries f, FormalPowerSeries g) {
if (f.size() < 60) return FPSNaive<mint>(f).div_mod(g).first;
f.cut_trailing_zeros(), g.cut_trailing_zeros();
const int fd = f.deg(), gd = g.deg();
assert(gd >= 0);
if (fd < gd) return {};
if (gd == 0) {
f /= g[0];
return f;
}
std::reverse(f.begin(), f.end()), std::reverse(g.begin(), g.end());
const int qd = fd - gd;
f.cut(qd + 1);
FormalPowerSeries q = f * g.inv(qd + 1);
q.cut(qd + 1);
std::reverse(q.begin(), q.end());
return q;
}
friend FormalPowerSeries operator%(const FormalPowerSeries& f, const FormalPowerSeries& g) { return f.div_mod(g).second; }
std::pair<FormalPowerSeries, FormalPowerSeries> div_mod(const FormalPowerSeries& g) const {
if (size() < 60) {
auto [q, r] = FPSNaive<mint>(*this).div_mod(g);
return { q, r };
}
FormalPowerSeries q = *this / g, r = *this - g * q;
r.cut_trailing_zeros();
return { q, r };
}
/* Shift Operations */
FormalPowerSeries& operator<<=(const int shamt) {
return this->insert(this->begin(), shamt, 0), * this;
}
FormalPowerSeries& operator>>=(const int shamt) {
return this->erase(this->begin(), this->begin() + std::min(shamt, size())), * this;
}
friend FormalPowerSeries operator<<(FormalPowerSeries f, const int shamt) { f <<= shamt; return f; }
friend FormalPowerSeries operator>>(FormalPowerSeries f, const int shamt) { f >>= shamt; return f; }
/* Compare */
friend bool operator==(const FormalPowerSeries& f, const FormalPowerSeries& g) {
const int n = f.size(), m = g.size();
if (n < m) return g == f;
for (int i = 0; i < m; ++i) if (f[i] != g[i]) return false;
for (int i = m; i < n; ++i) if (f[i] != 0) return false;
return true;
}
friend bool operator!=(const FormalPowerSeries& f, const FormalPowerSeries& g) { return not (f == g); }
/* Other Operations */
FormalPowerSeries& diff_inplace() {
if (this->empty()) return *this;
const int n = size();
for (int i = 1; i < n; ++i) (*this)[i - 1] = (*this)[i] * i;
return (*this)[n - 1] = 0, *this;
}
FormalPowerSeries diff() const {
FormalPowerSeries res = *this;
res.diff_inplace();
return res;
}
FormalPowerSeries& intg_inplace() {
const int n = size();
inv_mods<value_type> invs(n);
this->resize(n + 1);
for (int i = n; i > 0; --i) (*this)[i] = (*this)[i - 1] * invs[i];
return (*this)[0] = 0, *this;
}
FormalPowerSeries intg() const {
FormalPowerSeries res = *this;
res.intg_inplace();
return res;
}
FormalPowerSeries& inv_inplace(int n = -1) { return *this = inv(n); }
// reference: https://opt-cp.com/fps-fast-algorithms/
FormalPowerSeries inv(int n = -1) const {
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).inv();
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return inv_sparse(std::move(*sp_f), n);
FormalPowerSeries f_fft, g_fft;
FormalPowerSeries g{ (*this)[0].inv() };
for (int k = 1; k < n; k *= 2) {
f_fft = cut_copy(2 * k), g_fft = g.cut_copy(2 * k);
atcoder::internal::butterfly(f_fft);
atcoder::internal::butterfly(g_fft);
update_inv(k, f_fft, g_fft, g);
}
g.resize(n);
return g;
}
FormalPowerSeries& log_inplace(int n = -1) { return *this = log(n); }
FormalPowerSeries log(int n = -1) const {
assert(safe_get(0) == 1);
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).log();
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return log_sparse(std::move(*sp_f), n);
FormalPowerSeries res = inv(n) * diff();
res.resize(n - 1);
return res.intg();
}
FormalPowerSeries& exp_inplace(int n = -1) { return *this = exp(n); }
// https://arxiv.org/pdf/1301.5804.pdf
FormalPowerSeries exp(int n = -1) const {
assert(safe_get(0) == 0);
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).exp();
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return exp_sparse(std::move(*sp_f), n);
// h = *this
// f = exp(h) mod x ^ k
// g = f^{-1} mod x ^ k
FormalPowerSeries dh = diff();
FormalPowerSeries f{ 1 }, f_fft;
FormalPowerSeries g{ 1 }, g_fft;
for (int k = 1; k < n; k *= 2) {
f_fft = f.cut_copy(2 * k), atcoder::internal::butterfly(f_fft);
if (k > 1) update_inv(k / 2, f_fft, g_fft, g);
FormalPowerSeries t = f.cut_copy(k);
t.diff_inplace();
{
FormalPowerSeries r = dh.cut_copy(k);
r.back() = 0;
atcoder::internal::butterfly(r);
for (int i = 0; i < k; ++i) r[i] *= f_fft[i];
atcoder::internal::butterfly_inv(r);
r /= -k;
t += r;
t <<= 1, t[0] = t[k], t.pop_back();
}
t.resize(2 * k);
atcoder::internal::butterfly(t);
g_fft = g.cut_copy(2 * k);
atcoder::internal::butterfly(g_fft);
for (int i = 0; i < 2 * k; ++i) t[i] *= g_fft[i];
atcoder::internal::butterfly_inv(t);
t.resize(k);
t /= 2 * k;
FormalPowerSeries v = cut_copy(2 * k) >>= k;
t <<= k - 1;
t.intg_inplace();
for (int i = 0; i < k; ++i) v[i] -= t[k + i];
v.resize(2 * k);
atcoder::internal::butterfly(v);
for (int i = 0; i < 2 * k; ++i) v[i] *= f_fft[i];
atcoder::internal::butterfly_inv(v);
v.resize(k);
v /= 2 * k;
f.resize(2 * k);
for (int i = 0; i < k; ++i) f[k + i] = v[i];
}
f.cut(n);
return f;
}
FormalPowerSeries& pow_inplace(long long k, int n = -1) { return *this = pow(k, n); }
FormalPowerSeries pow(const long long k, int n = -1) const {
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).pow(k);
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return pow_sparse(std::move(*sp_f), k, n);
if (k == 0) {
FormalPowerSeries f{ 1 };
f.resize(n);
return f;
}
int tlz = 0;
while (tlz < size() and (*this)[tlz] == 0) ++tlz;
if (tlz == size() or tlz > (n - 1) / k) return FormalPowerSeries(n, 0);
const int m = n - tlz * k;
FormalPowerSeries f = *this >> tlz;
value_type base = f[0];
return ((((f /= base).log(m) *= k).exp(m) *= base.pow(k)) <<= (tlz * k));
}
std::optional<FormalPowerSeries> safe_sqrt(int n = -1) const {
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).safe_sqrt();
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return safe_sqrt_sparse(std::move(*sp_f), n);
int tlz = 0;
while (tlz < size() and (*this)[tlz] == 0) ++tlz;
if (tlz == size()) return FormalPowerSeries(n, 0);
if (tlz & 1) return std::nullopt;
const int m = n - tlz / 2;
FormalPowerSeries h(this->begin() + tlz, this->end());
auto q0 = ::safe_sqrt(h[0]);
if (not q0.has_value()) return std::nullopt;
FormalPowerSeries f{ *q0 }, f_fft, g{ q0->inv() }, g_fft;
for (int k = 1; k < m; k *= 2) {
f_fft = f.cut_copy(2 * k), atcoder::internal::butterfly(f_fft);
if (k > 1) update_inv(k / 2, f_fft, g_fft, g);
g_fft = g.cut_copy(2 * k);
atcoder::internal::butterfly(g_fft);
FormalPowerSeries h_fft = h.cut_copy(2 * k);
atcoder::internal::butterfly(h_fft);
for (int i = 0; i < 2 * k; ++i) h_fft[i] = (h_fft[i] - f_fft[i] * f_fft[i]) * g_fft[i];
atcoder::internal::butterfly_inv(h_fft);
f.resize(2 * k);
const value_type iz = value_type(4 * k).inv();
for (int i = 0; i < k; ++i) f[k + i] = h_fft[k + i] * iz;
}
f.resize(m), f <<= (tlz / 2);
return f;
}
FormalPowerSeries& sqrt_inplace(int n = -1) { return *this = sqrt(n); }
FormalPowerSeries sqrt(int n = -1) const {
return *safe_sqrt(n);
}
value_type eval(value_type x) const {
value_type y = 0;
for (int i = size() - 1; i >= 0; --i) y = y * x + (*this)[i];
return y;
}
static FormalPowerSeries prod(const std::vector<FormalPowerSeries>& fs) {
if (fs.empty()) return { 1 };
std::deque<FormalPowerSeries> dq(fs.begin(), fs.end());
std::sort(dq.begin(), dq.end(), [](auto& f, auto& g) { return f.size() < g.size(); });
while (dq.size() >= 2) {
dq.push_back(dq[0] * dq[1]);
dq.pop_front();
dq.pop_front();
}
return dq.front();
}
std::optional<std::vector<std::pair<int, value_type>>> sparse_fps_format(int max_size) const {
std::vector<std::pair<int, value_type>> res;
for (int i = 0; i <= deg() and int(res.size()) <= max_size; ++i) if (value_type v = (*this)[i]; v != 0) res.emplace_back(i, v);
if (int(res.size()) > max_size) return std::nullopt;
return res;
}
private:
static void update_inv(const int k, FormalPowerSeries& f_fft, FormalPowerSeries& g_fft, FormalPowerSeries& g) {
FormalPowerSeries fg(2 * k);
for (int i = 0; i < 2 * k; ++i) fg[i] = f_fft[i] * g_fft[i];
atcoder::internal::butterfly_inv(fg);
fg >>= k, fg.resize(2 * k);
atcoder::internal::butterfly(fg);
for (int i = 0; i < 2 * k; ++i) fg[i] *= g_fft[i];
atcoder::internal::butterfly_inv(fg);
const value_type iz = value_type(2 * k).inv(), c = -iz * iz;
g.resize(2 * k);
for (int i = 0; i < k; ++i) g[k + i] = fg[i] * c;
}
static FormalPowerSeries div_fps_sparse(const FormalPowerSeries& f, const std::vector<std::pair<int, value_type>>& g, int n) {
const int siz = g.size();
assert(siz and g[0].first == 0);
const value_type inv_g0 = g[0].second.inv();
FormalPowerSeries h(n);
for (int i = 0; i < n; ++i) {
value_type v = f.safe_get(i);
for (int idx = 1; idx < siz; ++idx) {
const auto& [j, gj] = g[idx];
if (j > i) break;
v -= gj * h[i - j];
}
h[i] = v * inv_g0;
}
return h;
}
static FormalPowerSeries inv_sparse(const std::vector<std::pair<int, value_type>>& g, const int n) {
return div_fps_sparse(FormalPowerSeries{ 1 }, g, n);
}
static FormalPowerSeries exp_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
const int siz = f.size();
assert(not siz or f[0].first != 0);
FormalPowerSeries g(n);
g[0] = 1;
inv_mods<value_type> invs(n);
for (int i = 1; i < n; ++i) {
value_type v = 0;
for (const auto& [j, fj] : f) {
if (j > i) break;
v += j * fj * g[i - j];
}
v *= invs[i];
g[i] = v;
}
return g;
}
static FormalPowerSeries log_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
const int siz = f.size();
assert(siz and f[0].first == 0 and f[0].second == 1);
FormalPowerSeries g(n);
for (int idx = 1; idx < siz; ++idx) {
const auto& [j, fj] = f[idx];
if (j >= n) break;
g[j] = j * fj;
}
inv_mods<value_type> invs(n);
for (int i = 1; i < n; ++i) {
value_type v = g[i];
for (int idx = 1; idx < siz; ++idx) {
const auto& [j, fj] = f[idx];
if (j > i) break;
v -= fj * g[i - j] * (i - j);
}
v *= invs[i];
g[i] = v;
}
return g;
}
static FormalPowerSeries pow_sparse(const std::vector<std::pair<int, value_type>>& f, const long long k, const int n) {
if (k == 0) {
FormalPowerSeries res(n, 0);
res[0] = 1;
return res;
}
const int siz = f.size();
if (not siz) return FormalPowerSeries(n, 0);
const int p = f[0].first;
if (p > (n - 1) / k) return FormalPowerSeries(n, 0);
const value_type inv_f0 = f[0].second.inv();
const int lz = p * k;
FormalPowerSeries g(n);
g[lz] = f[0].second.pow(k);
inv_mods<value_type> invs(n);
for (int i = 1; lz + i < n; ++i) {
value_type v = 0;
for (int idx = 1; idx < siz; ++idx) {
auto [j, fj] = f[idx];
j -= p;
if (j > i) break;
v += fj * g[lz + i - j] * (value_type(k) * j - (i - j));
}
v *= invs[i] * inv_f0;
g[lz + i] = v;
}
return g;
}
static std::optional<FormalPowerSeries> safe_sqrt_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
const int siz = f.size();
if (not siz) return FormalPowerSeries(n, 0);
const int p = f[0].first;
if (p % 2 == 1) return std::nullopt;
if (p / 2 >= n) return FormalPowerSeries(n, 0);
const value_type inv_f0 = f[0].second.inv();
const int lz = p / 2;
FormalPowerSeries g(n);
auto opt_g0 = ::safe_sqrt(f[0].second);
if (not opt_g0.has_value()) return std::nullopt;
g[lz] = *opt_g0;
value_type k = mint(2).inv();
inv_mods<value_type> invs(n);
for (int i = 1; lz + i < n; ++i) {
value_type v = 0;
for (int idx = 1; idx < siz; ++idx) {
auto [j, fj] = f[idx];
j -= p;
if (j > i) break;
v += fj * g[lz + i - j] * (k * j - (i - j));
}
v *= invs[i] * inv_f0;
g[lz + i] = v;
}
return g;
}
static FormalPowerSeries sqrt_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
return *safe_sqrt(f, n);
}
};
} // namespace suisen
template <typename mint>
suisen::FormalPowerSeries<mint> sqrt(suisen::FormalPowerSeries<mint> a) {
return a.sqrt();
}
template <typename mint>
suisen::FormalPowerSeries<mint> log(suisen::FormalPowerSeries<mint> a) {
return a.log();
}
template <typename mint>
suisen::FormalPowerSeries<mint> exp(suisen::FormalPowerSeries<mint> a) {
return a.exp();
}
template <typename mint, typename T>
suisen::FormalPowerSeries<mint> pow(suisen::FormalPowerSeries<mint> a, T b) {
return a.pow(b);
}
template <typename mint>
suisen::FormalPowerSeries<mint> inv(suisen::FormalPowerSeries<mint> a) {
return a.inv();
}
#line 7 "test/src/polynomial/rook_polynomial/dummy.test.cpp"
#line 9 "test/src/polynomial/rook_polynomial/dummy.test.cpp"
using mint = atcoder::modint998244353;
namespace atcoder {
std::istream& operator>>(std::istream& in, mint &a) {
long long e; in >> e; a = e;
return in;
}
std::ostream& operator<<(std::ostream& out, const mint &a) {
out << a.val();
return out;
}
} // namespace atcoder
using fps = suisen::FormalPowerSeries<mint>;
#include <map>
fps naive(const std::vector<int> &h) {
std::map<int, fps> pd;
pd[0] = { 1 };
for (int k : h) {
std::map<int, fps> dp;
for (auto [mask, f] : pd) {
const int nxt_mask = mask ^ ((mask >> k) << k);
dp[nxt_mask] += f;
for (int pos = 0; pos < k; ++pos) if (not ((nxt_mask >> pos) & 1)) {
dp[nxt_mask | (1 << pos)] += f << 1;
}
}
pd.swap(dp);
}
fps res(h.size() + 1);
for (auto [_, f] : pd) res += f;
return res;
}
#line 48 "test/src/polynomial/rook_polynomial/dummy.test.cpp"
void test() {
const std::vector<int> h { 6, 2, 5, 3, 2, 4, 5, 2, 3 };
const int n = h.size();
fps f = suisen::rook_polynomial_skyline_board<fps>(h);
fps g = naive(h);
assert(f == g);
}
int main() {
test();
std::cout << "Hello World" << std::endl;
return 0;
}