結果
| 問題 |
No.2137 Stairs of Permutation
|
| コンテスト | |
| ユーザー |
 hos.lyric
|
| 提出日時 | 2022-11-25 21:30:26 |
| 言語 | C++14 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 1,063 ms / 2,000 ms |
| コード長 | 8,359 bytes |
| コンパイル時間 | 1,487 ms |
| コンパイル使用メモリ | 117,872 KB |
| 実行使用メモリ | 42,496 KB |
| 最終ジャッジ日時 | 2024-10-02 04:04:25 |
| 合計ジャッジ時間 | 14,107 ms |
|
ジャッジサーバーID （参考情報） |
judge2 / judge1 |
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| ファイルパターン | 結果 |
|---|---|
| sample | AC * 2 |
| other | AC * 23 |
ソースコード
#include <cassert>
#include <cmath>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <algorithm>
#include <bitset>
#include <complex>
#include <deque>
#include <functional>
#include <iostream>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <sstream>
#include <string>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
using namespace std;
using Int = long long;
template <class T1, class T2> ostream &operator<<(ostream &os, const pair<T1, T2> &a) { return os << "(" << a.first << ", " << a.second << ")"; };
template <class T> ostream &operator<<(ostream &os, const vector<T> &as) { const int sz = as.size(); os << "["; for (int i = 0; i < sz; ++i) { if (i >= 256) { os << ", ..."; break; } if (i > 0) { os << ", "; } os << as[i]; } return os << "]"; }
template <class T> void pv(T a, T b) { for (T i = a; i != b; ++i) cerr << *i << " "; cerr << endl; }
template <class T> bool chmin(T &t, const T &f) { if (t > f) { t = f; return true; } return false; }
template <class T> bool chmax(T &t, const T &f) { if (t < f) { t = f; return true; } return false; }
////////////////////////////////////////////////////////////////////////////////
template <unsigned M_> struct ModInt {
static constexpr unsigned M = M_;
unsigned x;
constexpr ModInt() : x(0U) {}
constexpr ModInt(unsigned x_) : x(x_ % M) {}
constexpr ModInt(unsigned long long x_) : x(x_ % M) {}
constexpr ModInt(int x_) : x(((x_ %= static_cast<int>(M)) < 0) ? (x_ + static_cast<int>(M)) : x_) {}
constexpr ModInt(long long x_) : x(((x_ %= static_cast<long long>(M)) < 0) ? (x_ + static_cast<long long>(M)) : x_) {}
ModInt &operator+=(const ModInt &a) { x = ((x += a.x) >= M) ? (x - M) : x; return *this; }
ModInt &operator-=(const ModInt &a) { x = ((x -= a.x) >= M) ? (x + M) : x; return *this; }
ModInt &operator*=(const ModInt &a) { x = (static_cast<unsigned long long>(x) * a.x) % M; return *this; }
ModInt &operator/=(const ModInt &a) { return (*this *= a.inv()); }
ModInt pow(long long e) const {
if (e < 0) return inv().pow(-e);
ModInt a = *this, b = 1U; for (; e; e >>= 1) { if (e & 1) b *= a; a *= a; } return b;
}
ModInt inv() const {
unsigned a = M, b = x; int y = 0, z = 1;
for (; b; ) { const unsigned q = a / b; const unsigned c = a - q * b; a = b; b = c; const int w = y - static_cast<int>(q) * z; y = z; z = w; }
assert(a == 1U); return ModInt(y);
}
ModInt operator+() const { return *this; }
ModInt operator-() const { ModInt a; a.x = x ? (M - x) : 0U; return a; }
ModInt operator+(const ModInt &a) const { return (ModInt(*this) += a); }
ModInt operator-(const ModInt &a) const { return (ModInt(*this) -= a); }
ModInt operator*(const ModInt &a) const { return (ModInt(*this) *= a); }
ModInt operator/(const ModInt &a) const { return (ModInt(*this) /= a); }
template <class T> friend ModInt operator+(T a, const ModInt &b) { return (ModInt(a) += b); }
template <class T> friend ModInt operator-(T a, const ModInt &b) { return (ModInt(a) -= b); }
template <class T> friend ModInt operator*(T a, const ModInt &b) { return (ModInt(a) *= b); }
template <class T> friend ModInt operator/(T a, const ModInt &b) { return (ModInt(a) /= b); }
explicit operator bool() const { return x; }
bool operator==(const ModInt &a) const { return (x == a.x); }
bool operator!=(const ModInt &a) const { return (x != a.x); }
friend std::ostream &operator<<(std::ostream &os, const ModInt &a) { return os << a.x; }
};
////////////////////////////////////////////////////////////////////////////////
// 0 = \sum_{j=0}^d (\sum_{k=0}^e css[j][k] i^k) as[i - j] (d <= i < |as|)
template <unsigned M>
vector<vector<ModInt<M>>> findPRecurrence(const vector<ModInt<M>> &as, int e,
bool verbose = false) {
using Mint = ModInt<M>;
const int asLen = as.size();
// asLen - d >= (d + 1) (e + 1) - 1 (otherwise definitely dim Ker >= 2)
const int d0 = (asLen + 2) / (e + 2) - 1;
if (d0 < 0) {
if (verbose) {
fprintf(stderr, "[findPRecurrence] |as| >= e must hold.\n");
fflush(stderr);
}
return {};
}
const int m = asLen - d0, n = (d0 + 1) * (e + 1);
vector<vector<Mint>> bss(m, vector<Mint>(n, 0));
for (int i = d0; i < asLen; ++i) for (int j = 0; j <= d0; ++j) {
Mint pw = 1;
for (int k = 0; k <= e; ++k) {
bss[i - d0][j * (e + 1) + k] = pw * as[i - j];
pw *= i;
}
}
int r = 0;
vector<int> hs;
for (int h = 0; h < n; ++h) {
for (int i = r; i < m; ++i) if (bss[i][h]) {
bss[r].swap(bss[i]);
break;
}
if (r < m && bss[r][h]) {
const Mint s = bss[r][h].inv();
for (int j = h; j < n; ++j) bss[r][j] *= s;
for (int i = 0; i < m; ++i) if (i != r) {
const Mint t = bss[i][h];
for (int j = h; j < n; ++j) bss[i][j] -= t * bss[r][j];
}
++r;
} else {
hs.push_back(h);
}
}
if (hs.empty()) {
if (verbose) {
fprintf(stderr, "[findPRecurrence] Not found: d = %d, e = %d.\n", d0, e);
fflush(stderr);
}
return {};
}
vector<vector<Mint>> css(d0 + 1, vector<Mint>(e + 1, 0));
for (int j = 0; j <= d0; ++j) for (int k = 0; k <= e; ++k) {
const int h = j * (e + 1) + k;
css[j][k] = (h < hs[0]) ? -bss[h][hs[0]] : (h == hs[0]) ? 1 : 0;
}
int d = hs[0] / (e + 1);
for (int i = d0; i < asLen; ++i) {
Mint sum = 0;
for (int j = 0; j <= d0; ++j) {
Mint coef = 0, pw = 1;
for (int k = 0; k <= e; ++k) {
coef += css[j][k] * pw;
pw *= i;
}
sum += coef * as[i - j];
}
for (; sum; ) {
if (i - ++d < 0) break;
assert(d <= d0);
Mint coef = 0, pw = 1;
for (int k = 0; k <= e; ++k) {
coef += css[d][k] * pw;
pw *= i;
}
sum += coef * as[i - d];
}
}
css.resize(d + 1);
if (verbose) {
const int hsLen = hs.size();
if (hsLen > d0 - d + 1) {
fprintf(stderr, "[findPRecurrence] Degenerate? (dim Ker = %d)\n", hsLen);
}
fprintf(stderr, "[findPRecurrence] d = %d, e = %d, non-trivial: %d\n", d, e,
asLen - d - (d + 1) * (e + 1) + 1);
fprintf(stderr, "{\n");
for (int j = 0; j <= d; ++j) {
fprintf(stderr, " {");
for (int k = 0; k <= e; ++k) {
const unsigned c = css[j][k].x;
if (k > 0) fprintf(stderr, ", ");
fprintf(stderr, "%d", static_cast<int>((c < M - c) ? c : (c - M)));
}
fprintf(stderr, "},\n");
}
fprintf(stderr, "}\n");
fflush(stderr);
}
return css;
}
// 0 = \sum_{j=0}^d (\sum_{k=0}^e css[j][k] i^k) as[i - j] (d <= i < |as|)
template <unsigned M>
vector<ModInt<M>> extendPRecurrence(vector<ModInt<M>> as,
const vector<vector<ModInt<M>>> &css,
int n) {
using Mint = ModInt<M>;
assert(!css.empty());
const int d = css.size() - 1, e = css[0].size() - 1;
for (int j = 0; j <= d; ++j) assert(static_cast<int>(css[j].size()) == e + 1);
const int asLen = as.size();
as.resize(n);
for (int i = asLen; i < n; ++i) {
Mint sum = 0;
for (int j = 1; j <= d; ++j) {
Mint coef = 0, pw = 1;
for (int k = 0; k <= e; ++k) {
coef += css[j][k] * pw;
pw *= i;
}
sum += coef * as[i - j];
}
{
Mint coef = 0, pw = 1;
for (int k = 0; k <= e; ++k) {
coef += css[0][k] * pw;
pw *= i;
}
as[i] = -sum / coef;
}
}
return as;
}
////////////////////////////////////////////////////////////////////////////////
constexpr unsigned MO = 998244353;
using Mint = ModInt<MO>;
constexpr int M = 100;
Mint c[M][M];
int main() {
for (int n = 0; n < M; ++n) {
c[n][0] = 0;
c[n][n] = 1;
for (int k = 1; k < n; ++k) {
c[n][k] = c[n - 1][k - 1] + (n - 1) * c[n - 1][k];
}
}
vector<Mint> as(M);
for (int n = 0; n < M; ++n) {
for (int k = 0; k <= n; ++k) {
as[n] += c[n][k] * k * k * k;
}
}
cerr<<"as = "<<as<<endl;
/*
for (int e = 0; e <= 10; ++e) {
findPRecurrence(as, e, true);
}
*/
const auto css = findPRecurrence(as, 4, true);
int N;
for (; ~scanf("%d", &N); ) {
const auto res = extendPRecurrence(as, css, N + 1);
printf("%u\n", res[N].x);
}
return 0;
}