This documentation is automatically generated by online-judge-tools/verification-helper
#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;
}