cp-library-cpp

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:heavy_check_mark: Array Set Power Series
(library/math/array_set_power_series.hpp)

Array Set Power Series

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Code

#ifndef SUISEN_ARRAY_SPS
#define SUISEN_ARRAY_SPS

#include "library/convolution/array_subset_convolution.hpp"

namespace suisen {
    template <typename T, std::size_t N>
    struct ArraySetPowerSeries: public std::vector<T> {
        using base_type = std::vector<T>;
        using value_type = typename base_type::value_type;
        using size_type = typename base_type::size_type;

        using polynomial_type = array_ranked_subset_transform::polynomial_t<value_type, N>;

        using base_type::vector;

        ArraySetPowerSeries(): ArraySetPowerSeries(0) {}
        ArraySetPowerSeries(size_type n): ArraySetPowerSeries(n, value_type{ 0 }) {}
        ArraySetPowerSeries(size_type n, const value_type& val): ArraySetPowerSeries(std::vector<value_type>(1 << n, val)) {}
        ArraySetPowerSeries(const base_type& a): ArraySetPowerSeries(base_type(a)) {}
        ArraySetPowerSeries(base_type&& a): base_type(std::move(a)) {
            const int n = this->size();
            assert(n == (-n & n));
        }
        ArraySetPowerSeries(std::initializer_list<value_type> l): ArraySetPowerSeries(base_type(l)) {}

        static ArraySetPowerSeries one(int n) {
            ArraySetPowerSeries f(n, value_type{ 0 });
            f[0] = value_type{ 1 };
            return f;
        }

        void set_cardinality(int n) {
            this->resize(1 << n, value_type{ 0 });
        }
        int cardinality() const {
            return __builtin_ctz(this->size());
        }

        ArraySetPowerSeries cut_lower(size_type p) const {
            return ArraySetPowerSeries(this->begin(), this->begin() + p);
        }
        ArraySetPowerSeries cut_upper(size_type p) const {
            return ArraySetPowerSeries(this->begin() + p, this->begin() + p + p);
        }

        void concat(const ArraySetPowerSeries& upper) {
            assert(this->size() == upper.size());
            this->insert(this->end(), upper.begin(), upper.end());
        }

        ArraySetPowerSeries operator+() const {
            return *this;
        }
        ArraySetPowerSeries operator-() const {
            ArraySetPowerSeries res(*this);
            for (auto& e : res) e = -e;
            return res;
        }
        ArraySetPowerSeries& operator+=(const ArraySetPowerSeries& g) {
            for (size_type i = 0; i < g.size(); ++i) (*this)[i] += g[i];
            return *this;
        }
        ArraySetPowerSeries& operator-=(const ArraySetPowerSeries& g) {
            for (size_type i = 0; i < g.size(); ++i) (*this)[i] -= g[i];
            return *this;
        }
        ArraySetPowerSeries& operator*=(const ArraySetPowerSeries& g) {
            return *this = (zeta() *= g).mobius_inplace();
        }
        ArraySetPowerSeries& operator*=(const value_type& c) {
            for (auto& e : *this) e *= c;
            return *this;
        }
        ArraySetPowerSeries& operator/=(const value_type& c) {
            value_type inv_c = ::inv(c);
            for (auto& e : *this) e *= inv_c;
            return *this;
        }
        friend ArraySetPowerSeries operator+(ArraySetPowerSeries f, const ArraySetPowerSeries& g) { f += g; return f; }
        friend ArraySetPowerSeries operator-(ArraySetPowerSeries f, const ArraySetPowerSeries& g) { f -= g; return f; }
        friend ArraySetPowerSeries operator*(ArraySetPowerSeries f, const ArraySetPowerSeries& g) { f *= g; return f; }
        friend ArraySetPowerSeries operator*(ArraySetPowerSeries f, const value_type& c) { f *= c; return f; }
        friend ArraySetPowerSeries operator*(const value_type& c, ArraySetPowerSeries f) { f *= c; return f; }
        friend ArraySetPowerSeries operator/(ArraySetPowerSeries f, const value_type& c) { f /= c; return f; }

        ArraySetPowerSeries inv() {
            return zeta().inv_inplace().mobius_inplace();
        }
        ArraySetPowerSeries sqrt() {
            return zeta().sqrt_inplace().mobius_inplace();
        }
        ArraySetPowerSeries exp() {
            return zeta().exp_inplace().mobius_inplace();
        }
        ArraySetPowerSeries log() {
            return zeta().log_inplace().mobius_inplace();
        }
        ArraySetPowerSeries pow(long long k) {
            return zeta().pow_inplace(k).mobius_inplace();
        }

        struct ZetaSPS: public std::vector<polynomial_type> {
            using base_type = std::vector<polynomial_type>;
            ZetaSPS() = default;
            ZetaSPS(const ArraySetPowerSeries<value_type, N>& f): base_type::vector(array_ranked_subset_transform::ranked_zeta<T, N>(f)), _d(f.cardinality()) {}

            ZetaSPS operator+() const {
                return *this;
            }
            ZetaSPS operator-() const {
                ZetaSPS res(*this);
                for (auto& f : res) f = -f;
                return res;
            }
            friend ZetaSPS operator+(ZetaSPS f, const ZetaSPS& g) { f += g; return f; }
            friend ZetaSPS operator-(ZetaSPS f, const ZetaSPS& g) { f -= g; return f; }
            friend ZetaSPS operator*(ZetaSPS f, const ZetaSPS& g) { f *= g; return f; }
            friend ZetaSPS operator*(ZetaSPS f, const value_type& c) { f *= c; return f; }
            friend ZetaSPS operator*(const value_type& c, ZetaSPS f) { f *= c; return f; }
            friend ZetaSPS operator/(ZetaSPS f, const value_type& c) { f /= c; return f; }

            ZetaSPS& operator+=(const ZetaSPS& rhs) {
                assert(_d == rhs._d);
                for (int i = 0; i < 1 << _d; ++i) (*this)[i] += rhs[i];
                return *this;
            }
            ZetaSPS& operator-=(const ZetaSPS& rhs) {
                assert(_d == rhs._d);
                for (int i = 0; i < 1 << _d; ++i) (*this)[i] -= rhs[i];
                return *this;
            }
            ZetaSPS& operator*=(value_type c) {
                for (auto& f : *this) f *= c;
                return *this;
            }
            ZetaSPS& operator/=(value_type c) {
                value_type inv_c = ::inv(c);
                for (auto& f : *this) f *= inv_c;
                return *this;
            }
            ZetaSPS& operator*=(const ZetaSPS& rhs) {
                assert(_d == rhs._d);
                for (size_type i = 0; i < size_type(1) << _d; ++i) (*this)[i] = (*this)[i].mul(rhs[i]);
                return *this;
            }
            ZetaSPS inv()  const { auto f = ZetaSPS(*this).inv_inplace();  return f; }
            ZetaSPS sqrt() const { auto f = ZetaSPS(*this).sqrt_inplace(); return f; }
            ZetaSPS exp()  const { auto f = ZetaSPS(*this).exp_inplace();  return f; }
            ZetaSPS log()  const { auto f = ZetaSPS(*this).log_inplace();  return f; }
            ZetaSPS pow(long long k) const { auto f = ZetaSPS(*this).pow_inplace(k); return f; }
            ZetaSPS& inv_inplace() {
                for (auto& f : *this) f = f.inv();
                return *this;
            }
            ZetaSPS& sqrt_inplace() {
                for (auto& f : *this) f = f.sqrt();
                return *this;
            }
            ZetaSPS& exp_inplace() {
                for (auto& f : *this) f = f.exp();
                return *this;
            }
            ZetaSPS& log_inplace() {
                for (auto& f : *this) f = f.log();
                return *this;
            }
            ZetaSPS& pow_inplace(long long k) {
                for (auto& f : *this) f = f.pow(k);
                return *this;
            }
            ArraySetPowerSeries<value_type, N> mobius_inplace() {
                return array_ranked_subset_transform::deranked_mobius<value_type, N>(*this);
            }
            ArraySetPowerSeries<value_type, N> mobius() const {
                auto rf = ZetaSPS(*this);
                return array_ranked_subset_transform::deranked_mobius<value_type, N>(rf);
            }
        private:
            int _d;
        };

        ZetaSPS zeta() const {
            return ZetaSPS(*this);
        }
    };
} // namespace suisen

#endif // SUISEN_ARRAY_SPS
#line 1 "library/math/array_set_power_series.hpp"



#line 1 "library/convolution/array_subset_convolution.hpp"



#line 1 "library/polynomial/array_fps_naive.hpp"



#include <cassert>
#include <cmath>
#include <limits>
#include <type_traits>
#include <array>

#line 1 "library/type_traits/type_traits.hpp"



#line 5 "library/type_traits/type_traits.hpp"
#include <iostream>
#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/array_fps_naive.hpp"

#line 1 "library/math/modint_extension.hpp"



#line 5 "library/math/modint_extension.hpp"
#include <optional>

/**
 * reference: 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"



#include <vector>

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/array_fps_naive.hpp"

namespace suisen {
    template <typename T, std::size_t N>
    struct ArrayFPSNaive : std::array<T, N> {
        static constexpr int SIZE = N;
        static constexpr int DEG = SIZE - 1;

        using value_type = T;
        using element_type = rec_value_type_t<T>;

        ArrayFPSNaive() {
            this->fill(value_type{ 0 });
        }
        ArrayFPSNaive(const std::initializer_list<value_type> l) : ArrayFPSNaive() {
            std::copy(l.begin(), l.end(), this->begin());
        }

        ArrayFPSNaive operator+() const {
            return ArrayFPSNaive(*this);
        }
        ArrayFPSNaive operator-() const {
            ArrayFPSNaive f(*this);
            for (auto& e : f) e = -e;
            return f;
        }
        ArrayFPSNaive& operator++() { return ++(*this)[0], * this; }
        ArrayFPSNaive& operator--() { return --(*this)[0], * this; }
        ArrayFPSNaive& operator+=(const value_type x) { return (*this)[0] += x, *this; }
        ArrayFPSNaive& operator-=(const value_type x) { return (*this)[0] -= x, *this; }
        ArrayFPSNaive& operator+=(const ArrayFPSNaive& g) {
            for (int i = 0; i < SIZE; ++i) (*this)[i] += g[i];
            return *this;
        }
        ArrayFPSNaive& operator-=(const ArrayFPSNaive& g) {
            for (int i = 0; i < SIZE; ++i) (*this)[i] -= g[i];
            return *this;
        }
        ArrayFPSNaive& operator*=(const ArrayFPSNaive& g) { return *this = *this * g; }
        ArrayFPSNaive& operator*=(const value_type x) {
            for (auto& e : *this) e *= x;
            return *this;
        }
        ArrayFPSNaive& operator/=(const ArrayFPSNaive& g) { return *this = *this / g; }
        ArrayFPSNaive& operator%=(const ArrayFPSNaive& g) { return *this = *this % g; }
        ArrayFPSNaive& operator<<=(int shamt) {
            shamt = std::min(shamt, SIZE);
            for (int i = SIZE - 1; i >= shamt; --i) std::swap((*this)[i], (*this)[i - shamt]);
            std::fill(this->begin(), this->begin() + shamt, value_type{ 0 });
            return *this;
        }
        ArrayFPSNaive& operator>>=(int shamt) {
            shamt = std::min(shamt, SIZE);
            for (int i = 0; i < SIZE - shamt; ++i) std::swap((*this)[i], (*this)[i + shamt]);
            std::fill(this->begin() + (SIZE - shamt), this->end(), value_type{ 0 });
            return *this;
        }

        friend ArrayFPSNaive operator+(ArrayFPSNaive f, const ArrayFPSNaive& g) { f += g; return f; }
        friend ArrayFPSNaive operator+(ArrayFPSNaive f, const value_type& x) { f += x; return f; }
        friend ArrayFPSNaive operator-(ArrayFPSNaive f, const ArrayFPSNaive& g) { f -= g; return f; }
        friend ArrayFPSNaive operator-(ArrayFPSNaive f, const value_type& x) { f -= x; return f; }
        friend ArrayFPSNaive operator*(const ArrayFPSNaive& f, const ArrayFPSNaive& g) {
            ArrayFPSNaive h;
            for (int i = 0; i < SIZE; ++i) for (int j = 0; i + j < SIZE; ++j) h[i + j] += f[i] * g[j];
            return h;
        }
        friend ArrayFPSNaive operator*(ArrayFPSNaive f, const value_type& x) { f *= x; return f; }
        friend ArrayFPSNaive operator/(ArrayFPSNaive f, ArrayFPSNaive g) { return std::move(div_mod(std::move(f), std::move(g)).first); }
        friend ArrayFPSNaive operator%(ArrayFPSNaive f, ArrayFPSNaive g) { return std::move(div_mod(std::move(f), std::move(g)).second); }
        friend ArrayFPSNaive operator*(const value_type x, ArrayFPSNaive f) { f *= x; return f; }
        friend ArrayFPSNaive operator<<(ArrayFPSNaive f, const int shamt) { f <<= shamt; return f; }
        friend ArrayFPSNaive operator>>(ArrayFPSNaive f, const int shamt) { f >>= shamt; return f; }

        friend std::pair<ArrayFPSNaive, ArrayFPSNaive> div_mod(ArrayFPSNaive f, const ArrayFPSNaive& g) {
            int fd = DEG, gd = DEG;
            while (fd >= 0 and f[fd] == value_type{ 0 }) --fd;
            while (gd >= 0 and g[gd] == value_type{ 0 }) --gd;
            assert(gd >= 0);
            if (fd < gd) return { ArrayFPSNaive{}, f };
            if (gd == 0) return { f *= g[0].inv(), ArrayFPSNaive{} };
            const int k = fd - gd;
            value_type head_inv = g[gd].inv();
            ArrayFPSNaive q;
            for (int i = k; i >= 0; --i) {
                value_type div = f[i + gd] * head_inv;
                q[i] = div;
                for (int j = 0; j <= gd; ++j) f[i + j] -= div * g[j];
            }
            std::fill(f.begin() + gd, f.end(), value_type{ 0 });
            return { std::move(q), std::move(f) };
        }

        ArrayFPSNaive mul(const ArrayFPSNaive& g) const {
            return (*this) * g;
        }
        ArrayFPSNaive diff() const {
            ArrayFPSNaive g;
            for (int i = 1; i <= DEG; ++i) g[i - 1] = (*this)[i] * i;
            g[DEG] = 0;
            return g;
        }
        ArrayFPSNaive intg() const {
            ArrayFPSNaive g;
            for (int i = 0; i < DEG; ++i) g[i + 1] = (*this)[i] * invs[i + 1];
            return g;
        }
        ArrayFPSNaive inv() const {
            ArrayFPSNaive g;
            const value_type inv_f0 = ::inv((*this)[0]);
            g[0] = inv_f0;
            for (int i = 1; i <= DEG; ++i) {
                for (int j = 1; j <= i; ++j) g[i] -= g[i - j] * (*this)[j];
                g[i] *= inv_f0;
            }
            return g;
        }
        ArrayFPSNaive exp() const {
            assert((*this)[0] == value_type{ 0 });
            ArrayFPSNaive g;
            g[0] = value_type{ 1 };
            for (int i = 1; i <= DEG; ++i) {
                for (int j = 1; j <= i; ++j) g[i] += j * g[i - j] * (*this)[j];
                g[i] *= invs[i];
            }
            return g;
        }
        ArrayFPSNaive log() const {
            assert((*this)[0] == value_type{ 1 });
            ArrayFPSNaive g;
            g[0] = value_type{ 0 };
            for (int i = 1; i <= DEG; ++i) {
                g[i] = i * (*this)[i];
                for (int j = 1; j < i; ++j) g[i] -= (i - j) * g[i - j] * (*this)[j];
                g[i] *= invs[i];
            }
            return g;
        }
        ArrayFPSNaive pow(const long long k) const {
            if (k == 0) {
                ArrayFPSNaive g;
                g[0] = 1;
                return g;
            }
            int z = 0;
            while (z < SIZE and (*this)[z] == value_type{ 0 }) ++z;
            if (z >= DEG / k + 1) return ArrayFPSNaive{};
            const int d = DEG - z * k;
            const int bf = z, bg = z * k;

            ArrayFPSNaive g;
            const value_type inv_f0 = ::inv((*this)[bf]);
            g[bg] = (*this)[bf].pow(k);
            for (int i = 1; i <= d; ++i) {
                for (int j = 1; j <= i; ++j) g[bg + i] += (element_type{ k } * j - (i - j)) * g[bg + i - j] * (*this)[bf + j];
                g[bg + i] *= inv_f0 * invs[i];
            }
            return g;
        }

        ArrayFPSNaive sqrt() const {
            int dl = 0;
            while (dl < SIZE and (*this)[dl] == value_type{ 0 }) ++dl;
            if (dl == SIZE) return ArrayFPSNaive{};
            if (dl & 1) assert(false);

            const int d = DEG - dl / 2;
            const int bf = dl, bg = bf / 2;

            ArrayFPSNaive g;
            g[bg] = ::sqrt((*this)[bf]);
            value_type inv_2g0 = ::inv(2 * g[bg]);
            for (int i = 1; i <= d; ++i) {
                g[bg + i] = (*this)[bf + i];
                for (int j = 1; j < i; ++j) g[bg + i] -= g[bg + j] * g[bg + i - j];
                g[bg + i] *= inv_2g0;
            }
            return g;
        }

        value_type eval(value_type x) const {
            value_type y = 0;
            for (int i = DEG; i >= 0; --i) y = y * x + (*this)[i];
            return y;
        }

    private:
        static inline inv_mods<element_type> invs;
    };
} // namespace suisen

template <typename mint, std::size_t N>
auto sqrt(suisen::ArrayFPSNaive<mint, N> a) -> decltype(mint::mod(), suisen::ArrayFPSNaive<mint, N>{}) {
    return a.sqrt();
}
template <typename mint, std::size_t N>
auto log(suisen::ArrayFPSNaive<mint, N> a) -> decltype(mint::mod(), suisen::ArrayFPSNaive<mint, N>{}) {
    return a.log();
}
template <typename mint, std::size_t N>
auto exp(suisen::ArrayFPSNaive<mint, N> a) -> decltype(mint::mod(), suisen::ArrayFPSNaive<mint, N>{}) {
    return a.exp();
}
template <typename mint, std::size_t N, typename T>
auto pow(suisen::ArrayFPSNaive<mint, N> a, const T& b) -> decltype(mint::mod(), suisen::ArrayFPSNaive<mint, N>{}) {
    return a.pow(b);
}
template <typename mint, std::size_t N>
auto inv(suisen::ArrayFPSNaive<mint, N> a) -> decltype(mint::mod(), suisen::ArrayFPSNaive<mint, N>{}) {
    return a.inv();
}


#line 1 "library/transform/subset.hpp"



#line 1 "library/transform/kronecker_power.hpp"



#line 5 "library/transform/kronecker_power.hpp"
#include <utility>
#line 7 "library/transform/kronecker_power.hpp"

#line 1 "library/util/default_operator.hpp"



namespace suisen {
    namespace default_operator {
        template <typename T>
        auto zero() -> decltype(T { 0 }) { return T { 0 }; }
        template <typename T>
        auto one()  -> decltype(T { 1 }) { return T { 1 }; }
        template <typename T>
        auto add(const T &x, const T &y) -> decltype(x + y) { return x + y; }
        template <typename T>
        auto sub(const T &x, const T &y) -> decltype(x - y) { return x - y; }
        template <typename T>
        auto mul(const T &x, const T &y) -> decltype(x * y) { return x * y; }
        template <typename T>
        auto div(const T &x, const T &y) -> decltype(x / y) { return x / y; }
        template <typename T>
        auto mod(const T &x, const T &y) -> decltype(x % y) { return x % y; }
        template <typename T>
        auto neg(const T &x) -> decltype(-x) { return -x; }
        template <typename T>
        auto inv(const T &x) -> decltype(one<T>() / x)  { return one<T>() / x; }
    } // default_operator
    namespace default_operator_noref {
        template <typename T>
        auto zero() -> decltype(T { 0 }) { return T { 0 }; }
        template <typename T>
        auto one()  -> decltype(T { 1 }) { return T { 1 }; }
        template <typename T>
        auto add(T x, T y) -> decltype(x + y) { return x + y; }
        template <typename T>
        auto sub(T x, T y) -> decltype(x - y) { return x - y; }
        template <typename T>
        auto mul(T x, T y) -> decltype(x * y) { return x * y; }
        template <typename T>
        auto div(T x, T y) -> decltype(x / y) { return x / y; }
        template <typename T>
        auto mod(T x, T y) -> decltype(x % y) { return x % y; }
        template <typename T>
        auto neg(T x) -> decltype(-x) { return -x; }
        template <typename T>
        auto inv(T x) -> decltype(one<T>() / x)  { return one<T>() / x; }
    } // default_operator
} // namespace suisen


#line 9 "library/transform/kronecker_power.hpp"

namespace suisen {
    namespace kronecker_power_transform {
        namespace internal {
            template <typename UnitTransform, typename ReferenceGetter, std::size_t... Seq>
            void unit_transform(UnitTransform transform, ReferenceGetter ref_getter, std::index_sequence<Seq...>) {
                transform(ref_getter(Seq)...);
            }
        }

        template <typename T, std::size_t D, auto unit_transform>
        void kronecker_power_transform(std::vector<T> &x) {
            const std::size_t n = x.size();
            for (std::size_t block = 1; block < n; block *= D) {
                for (std::size_t l = 0; l < n; l += D * block) {
                    for (std::size_t offset = l; offset < l + block; ++offset) {
                        const auto ref_getter = [&](std::size_t i) -> T& { return x[offset + i * block]; };
                        internal::unit_transform(unit_transform, ref_getter, std::make_index_sequence<D>());
                    }
                }
            }
        }

        template <typename T, typename UnitTransform>
        void kronecker_power_transform(std::vector<T> &x, const std::size_t D, UnitTransform unit_transform) {
            const std::size_t n = x.size();
            std::vector<T> work(D);
            for (std::size_t block = 1; block < n; block *= D) {
                for (std::size_t l = 0; l < n; l += D * block) {
                    for (std::size_t offset = l; offset < l + block; ++offset) {
                        for (std::size_t i = 0; i < D; ++i) work[i] = x[offset + i * block];
                        unit_transform(work);
                        for (std::size_t i = 0; i < D; ++i) x[offset + i * block] = work[i];
                    }
                }
            }
        }

        template <typename T, auto e = default_operator::zero<T>, auto add = default_operator::add<T>, auto mul = default_operator::mul<T>>
        auto kronecker_power_transform(std::vector<T> &x, const std::vector<std::vector<T>> &A) -> decltype(e(), add(std::declval<T>(), std::declval<T>()), mul(std::declval<T>(), std::declval<T>()), void()) {
            const std::size_t D = A.size();
            assert(D == A[0].size());
            auto unit_transform = [&](std::vector<T> &x) {
                std::vector<T> y(D, e());
                for (std::size_t i = 0; i < D; ++i) for (std::size_t j = 0; j < D; ++j) {
                    y[i] = add(y[i], mul(A[i][j], x[j]));
                }
                x.swap(y);
            };
            kronecker_power_transform<T>(x, D, unit_transform);
        }
    }
} // namespace suisen



#line 5 "library/transform/subset.hpp"

namespace suisen::subset_transform {
    namespace internal {
        template <typename T, auto add = default_operator::add<T>>
        void zeta_unit_transform(T &x0, T &x1) {
                                // 1, 0
            x1 = add(x1, x0);   // 1, 1
        }
        template <typename T, auto sub = default_operator::sub<T>>
        void mobius_unit_transform(T &x0, T &x1) {
                                //  1, 0
            x1 = sub(x1, x0);   // -1, 1
        }
    } // namespace internal

    using kronecker_power_transform::kronecker_power_transform;

    template <typename T, auto add = default_operator::add<T>>
    void zeta(std::vector<T> &a) {
        kronecker_power_transform<T, 2, internal::zeta_unit_transform<T, add>>(a);
    }
    template <typename T, auto sub = default_operator::sub<T>>
    void mobius(std::vector<T> &a) {
        kronecker_power_transform<T, 2, internal::mobius_unit_transform<T, sub>>(a);
    }
} // namespace suisen::subset_transform


#line 6 "library/convolution/array_subset_convolution.hpp"

namespace suisen::array_ranked_subset_transform {
    template <typename T, std::size_t N>
    using polynomial_t = ArrayFPSNaive<T, N>;

    namespace internal {
        template <typename T, std::size_t N>
        std::vector<polynomial_t<T, N>> ranked(const std::vector<T>& a) {
            const int n = a.size();
            assert((-n & n) == n);
            std::vector fs(n, polynomial_t<T, N>{});
            for (int i = 0; i < n; ++i) fs[i][__builtin_popcount(i)] = a[i];
            return fs;
        }
        template <typename T, std::size_t N>
        std::vector<T> deranked(const std::vector<polynomial_t<T, N>>& polys) {
            const int n = polys.size();
            assert((-n & n) == n);
            std::vector<T> a(n);
            for (int i = 0; i < n; ++i) a[i] = polys[i][__builtin_popcount(i)];
            return a;
        }
    } // namespace suisen::array_ranked_subset_transform::internal

    template <typename T, std::size_t N>
    std::vector<polynomial_t<T, N>> ranked_zeta(const std::vector<T>& a) {
        auto ranked = internal::ranked<T, N>(a);
        subset_transform::zeta(ranked);
        return ranked;
    }
    template <typename T, std::size_t N>
    std::vector<T> deranked_mobius(std::vector<polynomial_t<T, N>>& ranked) {
        subset_transform::mobius(ranked);
        return internal::deranked<T, N>(ranked);
    }
} // namespace suisen::array_ranked_subset_transform


#line 5 "library/math/array_set_power_series.hpp"

namespace suisen {
    template <typename T, std::size_t N>
    struct ArraySetPowerSeries: public std::vector<T> {
        using base_type = std::vector<T>;
        using value_type = typename base_type::value_type;
        using size_type = typename base_type::size_type;

        using polynomial_type = array_ranked_subset_transform::polynomial_t<value_type, N>;

        using base_type::vector;

        ArraySetPowerSeries(): ArraySetPowerSeries(0) {}
        ArraySetPowerSeries(size_type n): ArraySetPowerSeries(n, value_type{ 0 }) {}
        ArraySetPowerSeries(size_type n, const value_type& val): ArraySetPowerSeries(std::vector<value_type>(1 << n, val)) {}
        ArraySetPowerSeries(const base_type& a): ArraySetPowerSeries(base_type(a)) {}
        ArraySetPowerSeries(base_type&& a): base_type(std::move(a)) {
            const int n = this->size();
            assert(n == (-n & n));
        }
        ArraySetPowerSeries(std::initializer_list<value_type> l): ArraySetPowerSeries(base_type(l)) {}

        static ArraySetPowerSeries one(int n) {
            ArraySetPowerSeries f(n, value_type{ 0 });
            f[0] = value_type{ 1 };
            return f;
        }

        void set_cardinality(int n) {
            this->resize(1 << n, value_type{ 0 });
        }
        int cardinality() const {
            return __builtin_ctz(this->size());
        }

        ArraySetPowerSeries cut_lower(size_type p) const {
            return ArraySetPowerSeries(this->begin(), this->begin() + p);
        }
        ArraySetPowerSeries cut_upper(size_type p) const {
            return ArraySetPowerSeries(this->begin() + p, this->begin() + p + p);
        }

        void concat(const ArraySetPowerSeries& upper) {
            assert(this->size() == upper.size());
            this->insert(this->end(), upper.begin(), upper.end());
        }

        ArraySetPowerSeries operator+() const {
            return *this;
        }
        ArraySetPowerSeries operator-() const {
            ArraySetPowerSeries res(*this);
            for (auto& e : res) e = -e;
            return res;
        }
        ArraySetPowerSeries& operator+=(const ArraySetPowerSeries& g) {
            for (size_type i = 0; i < g.size(); ++i) (*this)[i] += g[i];
            return *this;
        }
        ArraySetPowerSeries& operator-=(const ArraySetPowerSeries& g) {
            for (size_type i = 0; i < g.size(); ++i) (*this)[i] -= g[i];
            return *this;
        }
        ArraySetPowerSeries& operator*=(const ArraySetPowerSeries& g) {
            return *this = (zeta() *= g).mobius_inplace();
        }
        ArraySetPowerSeries& operator*=(const value_type& c) {
            for (auto& e : *this) e *= c;
            return *this;
        }
        ArraySetPowerSeries& operator/=(const value_type& c) {
            value_type inv_c = ::inv(c);
            for (auto& e : *this) e *= inv_c;
            return *this;
        }
        friend ArraySetPowerSeries operator+(ArraySetPowerSeries f, const ArraySetPowerSeries& g) { f += g; return f; }
        friend ArraySetPowerSeries operator-(ArraySetPowerSeries f, const ArraySetPowerSeries& g) { f -= g; return f; }
        friend ArraySetPowerSeries operator*(ArraySetPowerSeries f, const ArraySetPowerSeries& g) { f *= g; return f; }
        friend ArraySetPowerSeries operator*(ArraySetPowerSeries f, const value_type& c) { f *= c; return f; }
        friend ArraySetPowerSeries operator*(const value_type& c, ArraySetPowerSeries f) { f *= c; return f; }
        friend ArraySetPowerSeries operator/(ArraySetPowerSeries f, const value_type& c) { f /= c; return f; }

        ArraySetPowerSeries inv() {
            return zeta().inv_inplace().mobius_inplace();
        }
        ArraySetPowerSeries sqrt() {
            return zeta().sqrt_inplace().mobius_inplace();
        }
        ArraySetPowerSeries exp() {
            return zeta().exp_inplace().mobius_inplace();
        }
        ArraySetPowerSeries log() {
            return zeta().log_inplace().mobius_inplace();
        }
        ArraySetPowerSeries pow(long long k) {
            return zeta().pow_inplace(k).mobius_inplace();
        }

        struct ZetaSPS: public std::vector<polynomial_type> {
            using base_type = std::vector<polynomial_type>;
            ZetaSPS() = default;
            ZetaSPS(const ArraySetPowerSeries<value_type, N>& f): base_type::vector(array_ranked_subset_transform::ranked_zeta<T, N>(f)), _d(f.cardinality()) {}

            ZetaSPS operator+() const {
                return *this;
            }
            ZetaSPS operator-() const {
                ZetaSPS res(*this);
                for (auto& f : res) f = -f;
                return res;
            }
            friend ZetaSPS operator+(ZetaSPS f, const ZetaSPS& g) { f += g; return f; }
            friend ZetaSPS operator-(ZetaSPS f, const ZetaSPS& g) { f -= g; return f; }
            friend ZetaSPS operator*(ZetaSPS f, const ZetaSPS& g) { f *= g; return f; }
            friend ZetaSPS operator*(ZetaSPS f, const value_type& c) { f *= c; return f; }
            friend ZetaSPS operator*(const value_type& c, ZetaSPS f) { f *= c; return f; }
            friend ZetaSPS operator/(ZetaSPS f, const value_type& c) { f /= c; return f; }

            ZetaSPS& operator+=(const ZetaSPS& rhs) {
                assert(_d == rhs._d);
                for (int i = 0; i < 1 << _d; ++i) (*this)[i] += rhs[i];
                return *this;
            }
            ZetaSPS& operator-=(const ZetaSPS& rhs) {
                assert(_d == rhs._d);
                for (int i = 0; i < 1 << _d; ++i) (*this)[i] -= rhs[i];
                return *this;
            }
            ZetaSPS& operator*=(value_type c) {
                for (auto& f : *this) f *= c;
                return *this;
            }
            ZetaSPS& operator/=(value_type c) {
                value_type inv_c = ::inv(c);
                for (auto& f : *this) f *= inv_c;
                return *this;
            }
            ZetaSPS& operator*=(const ZetaSPS& rhs) {
                assert(_d == rhs._d);
                for (size_type i = 0; i < size_type(1) << _d; ++i) (*this)[i] = (*this)[i].mul(rhs[i]);
                return *this;
            }
            ZetaSPS inv()  const { auto f = ZetaSPS(*this).inv_inplace();  return f; }
            ZetaSPS sqrt() const { auto f = ZetaSPS(*this).sqrt_inplace(); return f; }
            ZetaSPS exp()  const { auto f = ZetaSPS(*this).exp_inplace();  return f; }
            ZetaSPS log()  const { auto f = ZetaSPS(*this).log_inplace();  return f; }
            ZetaSPS pow(long long k) const { auto f = ZetaSPS(*this).pow_inplace(k); return f; }
            ZetaSPS& inv_inplace() {
                for (auto& f : *this) f = f.inv();
                return *this;
            }
            ZetaSPS& sqrt_inplace() {
                for (auto& f : *this) f = f.sqrt();
                return *this;
            }
            ZetaSPS& exp_inplace() {
                for (auto& f : *this) f = f.exp();
                return *this;
            }
            ZetaSPS& log_inplace() {
                for (auto& f : *this) f = f.log();
                return *this;
            }
            ZetaSPS& pow_inplace(long long k) {
                for (auto& f : *this) f = f.pow(k);
                return *this;
            }
            ArraySetPowerSeries<value_type, N> mobius_inplace() {
                return array_ranked_subset_transform::deranked_mobius<value_type, N>(*this);
            }
            ArraySetPowerSeries<value_type, N> mobius() const {
                auto rf = ZetaSPS(*this);
                return array_ranked_subset_transform::deranked_mobius<value_type, N>(rf);
            }
        private:
            int _d;
        };

        ZetaSPS zeta() const {
            return ZetaSPS(*this);
        }
    };
} // namespace suisen
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