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#define PROBLEM "https://yukicoder.me/problems/no/2633" #include <atcoder/modint> #include "../../convolution/online_convolution.hpp" #include "../../math/binomial.hpp" #include "../../template/template.hpp" namespace ebi { using mint = atcoder::modint998244353; void main_() { const int max = 100'010; Binomial<mint> binom(max); int n; std::cin >> n; std::vector<int> a(n); std::cin >> a; std::vector<int> table(max, 0); rep(i, 0, n) table[a[i]]++; online_convolution oc; mint ans = 0; rep(i, 0, max) { mint ret = binom.inv_f(i); if (i > 0) ret += oc[i - 1]; ret *= mint(2).pow(table[i]) - 1; ans += ret * binom.f(i); oc.add(i, binom.inv_f(i + 1), ret); } std::cout << ans << '\n'; } } // namespace ebi int main() { ebi::fast_io(); int t = 1; // std::cin >> t; while (t--) { ebi::main_(); } return 0; }
#line 1 "test/yuki/yuki_2633.test.cpp" #define PROBLEM "https://yukicoder.me/problems/no/2633" #include <atcoder/modint> #line 2 "convolution/online_convolution.hpp" #include <atcoder/convolution> #line 5 "convolution/online_convolution.hpp" #include <bit> #include <vector> #line 2 "template/int_alias.hpp" #include <cstdint> namespace ebi { using ld = long double; using std::size_t; using i8 = std::int8_t; using u8 = std::uint8_t; using i16 = std::int16_t; using u16 = std::uint16_t; using i32 = std::int32_t; using u32 = std::uint32_t; using i64 = std::int64_t; using u64 = std::uint64_t; using i128 = __int128_t; using u128 = __uint128_t; } // namespace ebi #line 9 "convolution/online_convolution.hpp" namespace ebi { struct online_convolution { private: using mint = atcoder::modint998244353; public: online_convolution() = default; mint add(int idx, mint ai, mint bi) { assert(p == idx); a.emplace_back(ai); b.emplace_back(bi); int z = std::countr_zero(u32(p + 2)), w = 1 << z; if (p + 2 == w) { auto a0 = a; a0.resize(2 * w); atcoder::internal::butterfly(a0); fa.emplace_back(a0.begin(), a0.begin() + w); auto b0 = b; b0.resize(2 * w); atcoder::internal::butterfly(b0); fb.emplace_back(b0.begin(), b0.begin() + w); for (int i = 0; i < 2 * w; i++) a0[i] *= b0[i]; atcoder::internal::butterfly_inv(a0); mint inv_len = mint(2 * w).inv(); c.resize(2 * p + 2); for (int i = 0; i < p + 1; i++) c[p + i] += a0[p + i] * inv_len; } else { std::vector<mint> a0 = {a.end() - w, a.end()}; a0.resize(2 * w); atcoder::internal::butterfly(a0); std::vector<mint> b0 = {b.end() - w, b.end()}; b0.resize(2 * w); atcoder::internal::butterfly(b0); for (int i = 0; i < 2 * w; i++) { a0[i] = a0[i] * fb[z][i] + fa[z][i] * b0[i]; } atcoder::internal::butterfly_inv(a0); mint inv_len = mint(2 * w).inv(); for (int i = 0; i < w; i++) c[p + i] += a0[w - 1 + i] * inv_len; } return c[p++]; } mint operator[](int i) const { assert(0 <= i && i < p); return c[i]; } private: int p = 0; std::vector<mint> a, b, c; std::vector<std::vector<mint>> fa, fb; }; } // namespace ebi #line 2 "math/binomial.hpp" #line 4 "math/binomial.hpp" #include <cassert> #line 6 "math/binomial.hpp" #include <iostream> #include <ranges> #line 9 "math/binomial.hpp" #line 2 "modint/base.hpp" #include <concepts> #line 5 "modint/base.hpp" #include <utility> namespace ebi { template <class T> concept Modint = requires(T a, T b) { a + b; a - b; a * b; a / b; a.inv(); a.val(); a.pow(std::declval<long long>()); T::mod(); }; template <Modint mint> std::istream &operator>>(std::istream &os, mint &a) { long long x; os >> x; a = x; return os; } template <Modint mint> std::ostream &operator<<(std::ostream &os, const mint &a) { return os << a.val(); } } // namespace ebi #line 11 "math/binomial.hpp" namespace ebi { template <Modint mint> struct Binomial { private: static void extend(int len = -1) { int sz = (int)fact.size(); if (len < 0) len = 2 * sz; else if (len <= sz) return; else len = std::max(2 * sz, (int)std::bit_ceil(std::uint32_t(len))); len = std::min(len, mint::mod()); assert(sz <= len); fact.resize(len); inv_fact.resize(len); for (int i : std::views::iota(sz, len)) { fact[i] = fact[i - 1] * i; } inv_fact[len - 1] = fact[len - 1].inv(); for (int i : std::views::iota(sz, len) | std::views::reverse) { inv_fact[i - 1] = inv_fact[i] * i; } } public: Binomial() = default; Binomial(int n) { extend(n + 1); } static mint f(int n) { if (n >= (int)fact.size()) [[unlikely]] { extend(n + 1); } return fact[n]; } static mint inv_f(int n) { if (n >= (int)fact.size()) [[unlikely]] { extend(n + 1); } return inv_fact[n]; } static mint c(int n, int r) { if (r < 0 || n < r) return 0; return f(n) * inv_f(r) * inv_f(n - r); } static mint neg_c(int k, int d) { assert(d > 0); return c(k + d - 1, d - 1); } static mint p(int n, int r) { if (r < 0 || n < r) return 0; return f(n) * inv_f(n - r); } static mint catalan_number(int n) { return c(2 * n, n) * inv(n + 1); } static mint inv(int n) { return inv_f(n) * f(n - 1); } static void reserve(int n) { extend(n + 1); } private: static std::vector<mint> fact, inv_fact; }; template <Modint mint> std::vector<mint> Binomial<mint>::fact = std::vector<mint>(2, 1); template <Modint mint> std::vector<mint> Binomial<mint>::inv_fact = std::vector<mint>(2, 1); } // namespace ebi #line 1 "template/template.hpp" #include <bits/stdc++.h> #define rep(i, a, n) for (int i = (int)(a); i < (int)(n); i++) #define rrep(i, a, n) for (int i = ((int)(n)-1); i >= (int)(a); i--) #define Rep(i, a, n) for (i64 i = (i64)(a); i < (i64)(n); i++) #define RRep(i, a, n) for (i64 i = ((i64)(n)-i64(1)); i >= (i64)(a); i--) #define all(v) (v).begin(), (v).end() #define rall(v) (v).rbegin(), (v).rend() #line 2 "template/debug_template.hpp" #line 4 "template/debug_template.hpp" namespace ebi { #ifdef LOCAL #define debug(...) \ std::cerr << "LINE: " << __LINE__ << " [" << #__VA_ARGS__ << "]:", \ debug_out(__VA_ARGS__) #else #define debug(...) #endif void debug_out() { std::cerr << std::endl; } template <typename Head, typename... Tail> void debug_out(Head h, Tail... t) { std::cerr << " " << h; if (sizeof...(t) > 0) std::cerr << " :"; debug_out(t...); } } // namespace ebi #line 2 "template/io.hpp" #line 5 "template/io.hpp" #include <optional> #line 7 "template/io.hpp" namespace ebi { template <typename T1, typename T2> std::ostream &operator<<(std::ostream &os, const std::pair<T1, T2> &pa) { return os << pa.first << " " << pa.second; } template <typename T1, typename T2> std::istream &operator>>(std::istream &os, std::pair<T1, T2> &pa) { return os >> pa.first >> pa.second; } template <typename T> std::ostream &operator<<(std::ostream &os, const std::vector<T> &vec) { for (std::size_t i = 0; i < vec.size(); i++) os << vec[i] << (i + 1 == vec.size() ? "" : " "); return os; } template <typename T> std::istream &operator>>(std::istream &os, std::vector<T> &vec) { for (T &e : vec) std::cin >> e; return os; } template <typename T> std::ostream &operator<<(std::ostream &os, const std::optional<T> &opt) { if (opt) { os << opt.value(); } else { os << "invalid value"; } return os; } void fast_io() { std::cout << std::fixed << std::setprecision(15); std::cin.tie(nullptr); std::ios::sync_with_stdio(false); } } // namespace ebi #line 2 "template/utility.hpp" #line 5 "template/utility.hpp" #line 2 "graph/base.hpp" #line 7 "graph/base.hpp" #line 2 "data_structure/simple_csr.hpp" #line 6 "data_structure/simple_csr.hpp" namespace ebi { template <class E> struct simple_csr { simple_csr() = default; simple_csr(int n, const std::vector<std::pair<int, E>>& elements) : start(n + 1, 0), elist(elements.size()) { for (auto e : elements) { start[e.first + 1]++; } for (auto i : std::views::iota(0, n)) { start[i + 1] += start[i]; } auto counter = start; for (auto [i, e] : elements) { elist[counter[i]++] = e; } } simple_csr(const std::vector<std::vector<E>>& es) : start(es.size() + 1, 0) { int n = es.size(); for (auto i : std::views::iota(0, n)) { start[i + 1] = (int)es[i].size() + start[i]; } elist.resize(start.back()); for (auto i : std::views::iota(0, n)) { std::copy(es[i].begin(), es[i].end(), elist.begin() + start[i]); } } int size() const { return (int)start.size() - 1; } const auto operator[](int i) const { return std::ranges::subrange(elist.begin() + start[i], elist.begin() + start[i + 1]); } auto operator[](int i) { return std::ranges::subrange(elist.begin() + start[i], elist.begin() + start[i + 1]); } const auto operator()(int i, int l, int r) const { return std::ranges::subrange(elist.begin() + start[i] + l, elist.begin() + start[i + 1] + r); } auto operator()(int i, int l, int r) { return std::ranges::subrange(elist.begin() + start[i] + l, elist.begin() + start[i + 1] + r); } private: std::vector<int> start; std::vector<E> elist; }; } // namespace ebi #line 9 "graph/base.hpp" namespace ebi { template <class T> struct Edge { int from, to; T cost; int id; }; template <class E> struct Graph { using cost_type = E; using edge_type = Edge<cost_type>; Graph(int n_) : n(n_) {} Graph() = default; void add_edge(int u, int v, cost_type c) { buff.emplace_back(u, edge_type{u, v, c, m}); edges.emplace_back(edge_type{u, v, c, m++}); } void add_undirected_edge(int u, int v, cost_type c) { buff.emplace_back(u, edge_type{u, v, c, m}); buff.emplace_back(v, edge_type{v, u, c, m}); edges.emplace_back(edge_type{u, v, c, m}); m++; } void read_tree(int offset = 1, bool is_weighted = false) { read_graph(n - 1, offset, false, is_weighted); } void read_parents(int offset = 1) { for (auto i : std::views::iota(1, n)) { int p; std::cin >> p; p -= offset; add_undirected_edge(p, i, 1); } build(); } void read_graph(int e, int offset = 1, bool is_directed = false, bool is_weighted = false) { for (int i = 0; i < e; i++) { int u, v; std::cin >> u >> v; u -= offset; v -= offset; if (is_weighted) { cost_type c; std::cin >> c; if (is_directed) { add_edge(u, v, c); } else { add_undirected_edge(u, v, c); } } else { if (is_directed) { add_edge(u, v, 1); } else { add_undirected_edge(u, v, 1); } } } build(); } void build() { assert(!prepared); csr = simple_csr<edge_type>(n, buff); buff.clear(); prepared = true; } int size() const { return n; } int node_number() const { return n; } int edge_number() const { return m; } edge_type get_edge(int i) const { return edges[i]; } std::vector<edge_type> get_edges() const { return edges; } const auto operator[](int i) const { return csr[i]; } auto operator[](int i) { return csr[i]; } private: int n, m = 0; std::vector<std::pair<int,edge_type>> buff; std::vector<edge_type> edges; simple_csr<edge_type> csr; bool prepared = false; }; } // namespace ebi #line 8 "template/utility.hpp" namespace ebi { template <class T> inline bool chmin(T &a, T b) { if (a > b) { a = b; return true; } return false; } template <class T> inline bool chmax(T &a, T b) { if (a < b) { a = b; return true; } return false; } template <class T> T safe_ceil(T a, T b) { if (a % b == 0) return a / b; else if (a >= 0) return (a / b) + 1; else return -((-a) / b); } template <class T> T safe_floor(T a, T b) { if (a % b == 0) return a / b; else if (a >= 0) return a / b; else return -((-a) / b) - 1; } constexpr i64 LNF = std::numeric_limits<i64>::max() / 4; constexpr int INF = std::numeric_limits<int>::max() / 2; const std::vector<int> dy = {1, 0, -1, 0, 1, 1, -1, -1}; const std::vector<int> dx = {0, 1, 0, -1, 1, -1, 1, -1}; } // namespace ebi #line 8 "test/yuki/yuki_2633.test.cpp" namespace ebi { using mint = atcoder::modint998244353; void main_() { const int max = 100'010; Binomial<mint> binom(max); int n; std::cin >> n; std::vector<int> a(n); std::cin >> a; std::vector<int> table(max, 0); rep(i, 0, n) table[a[i]]++; online_convolution oc; mint ans = 0; rep(i, 0, max) { mint ret = binom.inv_f(i); if (i > 0) ret += oc[i - 1]; ret *= mint(2).pow(table[i]) - 1; ans += ret * binom.f(i); oc.add(i, binom.inv_f(i + 1), ret); } std::cout << ans << '\n'; } } // namespace ebi int main() { ebi::fast_io(); int t = 1; // std::cin >> t; while (t--) { ebi::main_(); } return 0; }