結果
| 問題 | No.1062 素敵なスコア |
| コンテスト | |
| ユーザー |
 ecottea
|
| 提出日時 | 2025-12-25 04:43:11 |
| 言語 | C++23 (gcc 13.3.0 + boost 1.89.0) |
| 結果 |
AC
|
| 実行時間 | 1,857 ms / 2,000 ms |
| コード長 | 45,942 bytes |
| 記録 | |
| コンパイル時間 | 8,771 ms |
| コンパイル使用メモリ | 366,060 KB |
| 実行使用メモリ | 7,848 KB |
| 最終ジャッジ日時 | 2025-12-25 04:44:25 |
| 合計ジャッジ時間 | 71,611 ms |
|
ジャッジサーバーID （参考情報） |
judge4 / judge1 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 32 |
ソースコード
#ifndef HIDDEN_IN_VS // 折りたたみ用
// 警告の抑制
#define _CRT_SECURE_NO_WARNINGS
// ライブラリの読み込み
#include <bits/stdc++.h>
using namespace std;
// 型名の短縮
using ll = long long; using ull = unsigned long long; // -2^63 ～ 2^63 = 9e18（int は -2^31 ～ 2^31 = 2e9）
using pii = pair<int, int>; using pll = pair<ll, ll>; using pil = pair<int, ll>; using pli = pair<ll, int>;
using vi = vector<int>; using vvi = vector<vi>; using vvvi = vector<vvi>; using vvvvi = vector<vvvi>;
using vl = vector<ll>; using vvl = vector<vl>; using vvvl = vector<vvl>; using vvvvl = vector<vvvl>;
using vb = vector<bool>; using vvb = vector<vb>; using vvvb = vector<vvb>;
using vc = vector<char>; using vvc = vector<vc>; using vvvc = vector<vvc>;
using vd = vector<double>; using vvd = vector<vd>; using vvvd = vector<vvd>;
template <class T> using priority_queue_rev = priority_queue<T, vector<T>, greater<T>>;
using Graph = vvi;
// 定数の定義
const double PI = acos(-1);
int DX[4] = { 1, 0, -1, 0 }; // 4 近傍（下，右，上，左）
int DY[4] = { 0, 1, 0, -1 };
int INF = 1001001001; ll INFL = 4004004003094073385LL; // (int)INFL = INF, (int)(-INFL) = -INF;
// 入出力高速化
struct fast_io { fast_io() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(18); } } fastIOtmp;
// 汎用マクロの定義
#define all(a) (a).begin(), (a).end()
#define sz(x) ((int)(x).size())
#define lbpos(a, x) (int)distance((a).begin(), std::lower_bound(all(a), (x)))
#define ubpos(a, x) (int)distance((a).begin(), std::upper_bound(all(a), (x)))
#define Yes(b) {cout << ((b) ? "Yes\n" : "No\n");}
#define rep(i, n) for(int i = 0, i##_len = int(n); i < i##_len; ++i) // 0 から n-1 まで昇順
#define repi(i, s, t) for(int i = int(s), i##_end = int(t); i <= i##_end; ++i) // s から t まで昇順
#define repir(i, s, t) for(int i = int(s), i##_end = int(t); i >= i##_end; --i) // s から t まで降順
#define repe(v, a) for(const auto& v : (a)) // a の全要素（変更不可能）
#define repea(v, a) for(auto& v : (a)) // a の全要素（変更可能）
#define repb(set, d) for(int set = 0, set##_ub = 1 << int(d); set < set##_ub; ++set) // d ビット全探索（昇順）
#define repis(i, set) for(int i = lsb(set), bset##i = set; i < 32; bset##i -= 1 << i, i = lsb(bset##i)) // set の全要素（昇順）
#define repp(a) sort(all(a)); for(bool a##_perm = true; a##_perm; a##_perm = next_permutation(all(a))) // a の順列全て（昇順）
#define uniq(a) {sort(all(a)); (a).erase(unique(all(a)), (a).end());} // 重複除去
#define EXIT(a) {cout << (a) << endl; exit(0);} // 強制終了
#define inQ(x, y, u, l, d, r) ((u) <= (x) && (l) <= (y) && (x) < (d) && (y) < (r)) // 半開矩形内判定
// 汎用関数の定義
template <class T> inline ll powi(T n, int k) { ll v = 1; rep(i, k) v *= n; return v; }
template <class T> inline bool chmax(T& M, const T& x) { if (M < x) { M = x; return true; } return false; } // 最大値を更新（更新されたら true を返す）
template <class T> inline bool chmin(T& m, const T& x) { if (m > x) { m = x; return true; } return false; } // 最小値を更新（更新されたら true を返す）
template <class T> inline int getb(T set, int i) { return (set >> i) & T(1); }
template <class T> inline T smod(T n, T m) { n %= m; if (n < 0) n += m; return n; } // 非負mod
// 演算子オーバーロード
template <class T, class U> inline istream& operator>>(istream& is, pair<T, U>& p) { is >> p.first >> p.second; return is; }
template <class T> inline istream& operator>>(istream& is, vector<T>& v) { repea(x, v) is >> x; return is; }
template <class T> inline vector<T>& operator--(vector<T>& v) { repea(x, v) --x; return v; }
template <class T> inline vector<T>& operator++(vector<T>& v) { repea(x, v) ++x; return v; }
#endif // 折りたたみ用
#if __has_include(<atcoder/all>)
#include <atcoder/all>
using namespace atcoder;
#ifdef _MSC_VER
#include "localACL.hpp"
#endif
using mint = modint998244353;
//using mint = static_modint<(int)1e9+7>;
//using mint = modint; // mint::set_mod(m);
using vm = vector<mint>; using vvm = vector<vm>; using vvvm = vector<vvm>; using vvvvm = vector<vvvm>; using pim = pair<int, mint>;
#endif
#ifdef _MSC_VER // 手元環境（Visual Studio）
#include "local.hpp"
#else // 提出用（gcc）
int mute_dump = 0;
int frac_print = 0;
#if __has_include(<atcoder/all>)
namespace atcoder {
inline istream& operator>>(istream& is, mint& x) { ll x_; is >> x_; x = x_; return is; }
inline ostream& operator<<(ostream& os, const mint& x) { os << x.val(); return os; }
}
#endif
inline int popcount(int n) { return __builtin_popcount(n); }
inline int popcount(ll n) { return __builtin_popcountll(n); }
inline int lsb(int n) { return n != 0 ? __builtin_ctz(n) : 32; }
inline int lsb(ll n) { return n != 0 ? __builtin_ctzll(n) : 64; }
inline int msb(int n) { return n != 0 ? (31 - __builtin_clz(n)) : -1; }
inline int msb(ll n) { return n != 0 ? (63 - __builtin_clzll(n)) : -1; }
#define dump(...)
#define dumpel(v)
#define dump_math(v)
#define input_from_file(f)
#define output_to_file(f)
#define Assert(b) { if (!(b)) { vc MLE(1<<30); rep(i,9)cout<<MLE[i]; exit(0); } } // RE の代わりに MLE を出す
#endif
//【階乗など（法が大きな素数）】
/*
* Factorial_mint(int N) : O(n)
* N まで計算可能として初期化する．
*
* mint fact(int n) : O(1)
* n! を返す．
*
* mint fact_inv(int n) : O(1)
* 1/n! を返す（n が負なら 0 を返す）
*
* mint inv(int n) : O(1)
* 1/n を返す．
*
* mint inv_neg(int n) : O(1)
* 1/n を返す（n < 0 も可）
*
* mint perm(int n, int r) : O(1)
* 順列の数 nPr を返す．
*
* mint perm_inv(int n, int r) : O(1)
* 順列の数の逆数 1/nPr を返す．
*
* mint bin(int n, int r) : O(1)
* 二項係数 nCr を返す．
*
* mint bin_inv(int n, int r) : O(1)
* 二項係数の逆数 1/nCr を返す．
*
* mint mul(vi rs) : O(|rs|)
* 多項係数 nC[rs] を返す．（n = Σrs）
*
* mint hom(int n, int r) : O(1)
* 重複組合せの数 nHr = n+r-1Cr を返す（0H0 = 1 とする）
*
* mint neg_bin(int n, int r) : O(1)
* 負の二項係数 nCr = (-1)^r -n+r-1Cr を返す（n ≦ 0, r ≧ 0）
*
* mint pochhammer(int x, int n) : O(1)
* ポッホハマー記号 x^(n) を返す（n ≧ 0）
*
* mint pochhammer_inv(int x, int n) : O(1)
* ポッホハマー記号の逆数 1/x^(n) を返す（n ≧ 0）
*/
class Factorial_mint {
int n_max;
// 階乗と階乗の逆数の値を保持するテーブル
vm fac, fac_inv;
public:
// n! までの階乗とその逆数を前計算しておく．O(n)
Factorial_mint(int n) : n_max(n), fac(n + 1), fac_inv(n + 1) {
// verify : https://atcoder.jp/contests/dwacon6th-prelims/tasks/dwacon6th_prelims_b
fac[0] = 1;
repi(i, 1, n) fac[i] = fac[i - 1] * i;
fac_inv[n] = fac[n].inv();
repir(i, n - 1, 0) fac_inv[i] = fac_inv[i + 1] * (i + 1);
}
Factorial_mint() : n_max(0) {} // ダミー
// n! を返す．
mint fact(int n) const {
// verify : https://atcoder.jp/contests/dwacon6th-prelims/tasks/dwacon6th_prelims_b
Assert(0 <= n && n <= n_max);
return fac[n];
}
// 1/n! を返す（n が負なら 0 を返す）
mint fact_inv(int n) const {
// verify : https://atcoder.jp/contests/abc289/tasks/abc289_h
Assert(n <= n_max);
if (n < 0) return 0;
return fac_inv[n];
}
// 1/n を返す．
mint inv(int n) const {
// verify : https://atcoder.jp/contests/exawizards2019/tasks/exawizards2019_d
Assert(n > 0);
Assert(n <= n_max);
return fac[n - 1] * fac_inv[n];
}
// 1/n を返す（n < 0 も可）
mint inv_neg(int n) const {
Assert(n != 0);
Assert(abs(n) <= n_max);
if (n > 0) return fac[n - 1] * fac_inv[n];
else return -fac[-n - 1] * fac_inv[-n];
}
// 順列の数 nPr を返す．
mint perm(int n, int r) const {
// verify : https://atcoder.jp/contests/abc172/tasks/abc172_e
Assert(n <= n_max);
if (r < 0 || n - r < 0) return 0;
return fac[n] * fac_inv[n - r];
}
// 順列の数 nPr の逆数を返す．
mint perm_inv(int n, int r) const {
// verify : https://yukicoder.me/problems/no/3139
Assert(n <= n_max);
Assert(0 <= r); Assert(r <= n);
return fac_inv[n] * fac[n - r];
}
// 二項係数 nCr を返す．
mint bin(int n, int r) const {
// verify : https://judge.yosupo.jp/problem/binomial_coefficient_prime_mod
Assert(n <= n_max);
if (r < 0 || n - r < 0) return 0;
return fac[n] * fac_inv[r] * fac_inv[n - r];
}
// 二項係数の逆数 1/nCr を返す．
mint bin_inv(int n, int r) const {
// verify : https://www.codechef.com/problems/RANDCOLORING
Assert(n <= n_max);
Assert(r >= 0);
Assert(n - r >= 0);
return fac_inv[n] * fac[r] * fac[n - r];
}
// 多項係数 nC[rs] を返す．
mint mul(const vi& rs) const {
// verify : https://yukicoder.me/problems/no/2141
if (rs.empty()) return 1;
if (*min_element(all(rs)) < 0) return 0;
int n = accumulate(all(rs), 0);
Assert(n <= n_max);
mint res = fac[n];
repe(r, rs) res *= fac_inv[r];
return res;
}
// 重複組合せの数 nHr = n+r-1Cr を返す（0H0 = 1 とする）
mint hom(int n, int r) {
// verify : https://mojacoder.app/users/riantkb/problems/toj_ex_2
if (n == 0) return (int)(r == 0);
if (r < 0 || n - 1 < 0) return 0;
Assert(n + r - 1 <= n_max);
return fac[n + r - 1] * fac_inv[r] * fac_inv[n - 1];
}
// 負の二項係数 nCr を返す（n ≦ 0, r ≧ 0）
mint neg_bin(int n, int r) {
// verify : https://atcoder.jp/contests/abc345/tasks/abc345_g
if (n == 0) return (int)(r == 0);
if (r < 0 || -n - 1 < 0) return 0;
Assert(-n + r - 1 <= n_max);
return (r & 1 ? -1 : 1) * fac[-n + r - 1] * fac_inv[r] * fac_inv[-n - 1];
}
// ポッホハマー記号 x^(n) を返す（n ≧ 0）
mint pochhammer(int x, int n) {
// verify : https://atcoder.jp/contests/agc070/tasks/agc070_c
int x2 = x + n - 1;
if (x <= 0 && 0 <= x2) return 0;
if (x > 0) {
Assert(x2 <= n_max);
return fac[x2] * fac_inv[x - 1];
}
else {
Assert(-x <= n_max);
return (n & 1 ? -1 : 1) * fac[-x] * fac_inv[-x2 - 1];
}
}
// ポッホハマー記号の逆数 1/x^(n) を返す（n ≧ 0）
mint pochhammer_inv(int x, int n) {
// verify : https://atcoder.jp/contests/agc070/tasks/agc070_c
int x2 = x + n - 1;
Assert(!(x <= 0 && 0 <= x2));
if (x > 0) {
Assert(x2 <= n_max);
return fac_inv[x2] * fac[x - 1];
}
else {
Assert(-x <= n_max);
return (n & 1 ? -1 : 1) * fac_inv[-x] * fac[-x2 - 1];
}
}
};
Factorial_mint fm((int)1e5 + 10);
// (i1, i2, i3) = (n1, n2, n3) に対する愚直解を返す．
mint naive_123(int i0, int i1, int i2) {
int a1 = i0 + 1;
int a2 = a1 + i1 + 1;
int n = a2 + i2 + 1;
// とりあえず多項式オーダーにした
int cL = a1;
int cM = a2 - a1;
int cR = n - a2;
//dump(cL, cM, cR);
mint res = 0;
//dump("a2 より大，不動 :");
repi(i, 1, n) {
repi(cLl, 0, cL) repi(cMl, 0, cM) {
int cLr = cL - cLl;
int cMr = cM - cMl;
int cRl = (i - 1) - cLl - cMl;
int cRr = (n - i) - cLr - cMr;
if (cRl < 0 || cRr < 0) continue;
if (cMl + cRl < cLr) continue;
if (cRl < cLr + cMr) continue;
mint pos = fm.mul({ cLl, cMl, cRl }) * fm.mul({ cLr, cMr, cRr });
mint val = fm.fact(cL) * fm.fact(cM) * fm.fact(cR - 1);
//dump("i:", i, "c:", cLl, cMl, cRl, cLr, cMr, cRr, ":", pos, val);
res += pos * val * cR;
}
}
//dump("a2 より大，移動 :");
repi(i, 1, n) repi(j, i + 1, n) {
repi(cLl, 0, cL) {
int cMl = 0;
int cRl = (i - 1) - cLl - cMl;
int cLr = cRl;
int cMr = 0;
int cRr = (n - j) - cLr - cMr;
int cLm = (cL - 1) - cLl - cLr;
int cMm = cM - cMl - cMr;
int cRm = (cR - 1) - cRl - cRr;
if (cRl < 0 || cRr < 0 || cLm < 0 || cMm < 0 || cRm < 0) continue;
mint pos = fm.mul({ cLl, cMl, cRl }) * fm.mul({ cLm, cMm, cRm }) * fm.mul({ cLr, cMr, cRr });
mint val = fm.fact(cL - 1) * fm.fact(cM) * fm.fact(cR - 1);
//dump("i:", i, "j:", j, "c:", cLl, cMl, cRl, cLm, cMm, cRm, cLr, cMr, cRr, ":", pos, val);
res += pos * val * cL * cR;
}
}
//dump("a1 より大，a2 以下，不動 :");
repi(i, 1, n) {
repi(cLl, 0, cL) repi(cMl, 0, cM - 1) {
int cLr = cL - cLl;
int cMr = (cM - 1) - cMl;
int cRl = (i - 1) - cLl - cMl;
int cRr = (n - i) - cLr - cMr;
if (cRl < 0 || cRr < 0) continue;
if (cMl + cRl < cLr) continue;
if (cRl > cLr + cMr) continue;
mint pos = fm.mul({ cLl, cMl, cRl }) * fm.mul({ cLr, cMr, cRr });
mint val = fm.fact(cL) * fm.fact(cM - 1) * fm.fact(cR);
//dump("i:", i, "c:", cLl, cMl, cRl, cLr, cMr, cRr, ":", pos, val);
res += pos * val * cM;
}
}
{
swap(a1, a2);
a1 = n - a1;
a2 = n - a2;
//dump(a1, a2);
cL = a1;
cM = a2 - a1;
cR = n - a2;
//dump("a2 より大，不動 :");
repi(i, 1, n) {
repi(cLl, 0, cL) repi(cMl, 0, cM) {
int cLr = cL - cLl;
int cMr = cM - cMl;
int cRl = (i - 1) - cLl - cMl;
int cRr = (n - i) - cLr - cMr;
if (cRl < 0 || cRr < 0) continue;
if (cMl + cRl < cLr) continue;
if (cRl < cLr + cMr) continue;
mint pos = fm.mul({ cLl, cMl, cRl }) * fm.mul({ cLr, cMr, cRr });
mint val = fm.fact(cL) * fm.fact(cM) * fm.fact(cR - 1);
//dump("i:", i, "c:", cLl, cMl, cRl, cLr, cMr, cRr, ":", pos, val);
res += pos * val * cR;
}
}
//dump("a2 より大，移動 :");
repi(i, 1, n) repi(j, i + 1, n) {
repi(cLl, 0, cL) {
int cMl = 0;
int cRl = (i - 1) - cLl - cMl;
int cLr = cRl;
int cMr = 0;
int cRr = (n - j) - cLr - cMr;
int cLm = (cL - 1) - cLl - cLr;
int cMm = cM - cMl - cMr;
int cRm = (cR - 1) - cRl - cRr;
if (cRl < 0 || cRr < 0 || cLm < 0 || cMm < 0 || cRm < 0) continue;
mint pos = fm.mul({ cLl, cMl, cRl }) * fm.mul({ cLm, cMm, cRm }) * fm.mul({ cLr, cMr, cRr });
mint val = fm.fact(cL - 1) * fm.fact(cM) * fm.fact(cR - 1);
//dump("i:", i, "j:", j, "c:", cLl, cMl, cRl, cLm, cMm, cRm, cLr, cMr, cRr, ":", pos, val);
res += pos * val * cL * cR;
}
}
}
return res;
}
// (i1, i2) = (n1, n2) に対する愚直解を返す．
vm naive_12(int n1, int n2) {
vm seq;
return seq;
}
// i1 = n1 に対する愚直解を返す．
vvm naive_1(int n1) {
vvm tbl;
return tbl;
}
// (i1,i2,i3)∈[0..n1)×[0..n2)×[0..n3) に対する愚直解を返す．
vvvm naive() {
int N1 = 14, N2 = 14, N3 = 14; // 15 だと TLE した
vvvm box;
rep(i1, N1) {
vvm tbl;
rep(i2, N2) {
vm seq;
rep(i3, N3) {
seq.push_back(naive_123(i1, i2, i3));
}
tbl.push_back(seq);
}
box.push_back(tbl);
}
//vvvm box(N1, vvm(N2, vm(N3)));
//rep(i1, N1) {
// dump("i1:", i1);
// rep(i2, N2) rep(i3, N3) {
// box[i1][i2][i3] = naive_123(i1, i2, i3);
// }
//}
//vvvm box(N1, vvm(N2));
//rep(i1, N1) {
// dump("i1:", i1);
// rep(i2, N2) {
// box[i1][i2] = naive_12(i1, i2);
// }
//}
//vvvm box(N1);
//rep(i1, N1) {
// dump("i1:", i1);
// box[i1] = naive_1(i1);
//}
#ifdef _MSC_VER
// 埋め込み用
string eb;
eb += "vvvm box = {\n";
rep(i1, sz(box)) {
eb += "{";
rep(i2, sz(box[i1])) {
eb += "{";
rep(i3, sz(box[i1][i2])) {
eb += to_string(box[i1][i2][i3].val()) + ",";
}
if (eb.back() == ',') eb.pop_back();
eb += "},";
}
if (eb.back() == ',') eb.pop_back();
eb += "},\n";
}
eb.pop_back(); eb.pop_back();
eb += "};\n\n";
cout << eb;
#endif
return box;
}
//【行列】
template <class T>
struct Matrix {
int n, m; // 行列のサイズ（n 行 m 列）
vector<vector<T>> v; // 行列の成分
// n×m 零行列で初期化する．
Matrix(int n, int m) : n(n), m(m), v(n, vector<T>(m)) {}
// n×n 単位行列で初期化する．
Matrix(int n) : n(n), m(n), v(n, vector<T>(n)) { rep(i, n) v[i][i] = T(1); }
// 二次元配列 a[0..n)[0..m) の要素で初期化する．
Matrix(const vector<vector<T>>& a) : n(sz(a)), m(sz(a[0])), v(a) {}
Matrix() : n(0), m(0) {}
// 代入
Matrix(const Matrix&) = default;
Matrix& operator=(const Matrix&) = default;
// アクセス
inline vector<T> const& operator[](int i) const { return v[i]; }
inline vector<T>& operator[](int i) {return v[i];}
// 入力
friend istream& operator>>(istream& is, Matrix& a) {
rep(i, a.n) rep(j, a.m) is >> a.v[i][j];
return is;
}
// 行の追加
void push_back(const vector<T>& a) {
Assert(sz(a) == m);
v.push_back(a);
n++;
}
// 行の削除
void pop_back() {
Assert(n > 0);
v.pop_back();
n--;
}
// サイズ変更
void resize(int n_) {
v.resize(n_);
n = n_;
}
void resize(int n_, int m_) {
n = n_;
m = m_;
v.resize(n);
rep(i, n) v[i].resize(m);
}
// 空か
bool empty() const { return min(n, m) == 0; }
// 比較
bool operator==(const Matrix& b) const { return n == b.n && m == b.m && v == b.v; }
bool operator!=(const Matrix& b) const { return !(*this == b); }
// 加算，減算，スカラー倍
Matrix& operator+=(const Matrix& b) {
rep(i, n) rep(j, m) v[i][j] += b[i][j];
return *this;
}
Matrix& operator-=(const Matrix& b) {
rep(i, n) rep(j, m) v[i][j] -= b[i][j];
return *this;
}
Matrix& operator*=(const T& c) {
rep(i, n) rep(j, m) v[i][j] *= c;
return *this;
}
Matrix operator+(const Matrix& b) const { return Matrix(*this) += b; }
Matrix operator-(const Matrix& b) const { return Matrix(*this) -= b; }
Matrix operator*(const T& c) const { return Matrix(*this) *= c; }
friend Matrix operator*(const T& c, const Matrix<T>& a) { return a * c; }
Matrix operator-() const { return Matrix(*this) *= T(-1); }
// 行列ベクトル積 : O(m n)
vector<T> operator*(const vector<T>& x) const {
vector<T> y(n);
rep(i, n) rep(j, m) y[i] += v[i][j] * x[j];
return y;
}
// ベクトル行列積 : O(m n)
friend vector<T> operator*(const vector<T>& x, const Matrix& a) {
vector<T> y(a.m);
rep(i, a.n) rep(j, a.m) y[j] += x[i] * a[i][j];
return y;
}
// 積：O(n^3)
Matrix operator*(const Matrix& b) const {
Matrix res(n, b.m);
rep(i, res.n) rep(k, m) rep(j, res.m) res[i][j] += v[i][k] * b[k][j];
return res;
}
Matrix& operator*=(const Matrix& b) { *this = *this * b; return *this; }
// 累乗：O(n^3 log d)
Matrix pow(ll d) const {
Matrix res(n), pow2 = *this;
while (d > 0) {
if (d & 1) res *= pow2;
pow2 *= pow2;
d >>= 1;
}
return res;
}
#ifdef _MSC_VER
friend ostream& operator<<(ostream& os, const Matrix& a) {
rep(i, a.n) {
os << "[";
rep(j, a.m) os << a[i][j] << " ]"[j == a.m - 1];
if (i < a.n - 1) os << "\n";
}
return os;
}
#endif
};
//【線形方程式】O(n m min(n, m))
template <class T>
vector<T> gauss_jordan_elimination(const Matrix<T>& A, const vector<T>& b, vector<vector<T>>* xs = nullptr) {
int n = A.n, m = A.m;
// v : 拡大係数行列 (A | b)
vector<vector<T>> v(n, vector<T>(m + 1));
rep(i, n) rep(j, m) v[i][j] = A[i][j];
rep(i, n) v[i][m] = b[i];
// pivots[i] : 第 i 行のピボットが第何列にあるか
vi pivots;
// 注目位置を v[i][j] とする．
int i = 0, j = 0;
while (i < n && j <= m) {
// 注目列の下方の行から非 0 成分を見つける．
int i2 = i;
while (i2 < n && v[i2][j] == T(0)) i2++;
// 見つからなかったら注目位置を右に移す．
if (i2 == n) { j++; continue; }
// 見つかったら第 i 行とその行を入れ替える．
if (i != i2) swap(v[i], v[i2]);
// v[i][j] をピボットに選択する．
pivots.push_back(j);
// v[i][j] が 1 になるよう第 i 行全体を v[i][j] で割る．
T vij_inv = T(1) / v[i][j];
repi(j2, j, m) v[i][j2] *= vij_inv;
// 第 i 行以外の第 j 列の成分が全て 0 になるよう第 i 行を定数倍して減じる．
rep(i2, n) {
if (v[i2][j] == T(0) || i2 == i) continue;
T mul = v[i2][j];
repi(j2, j, m) v[i2][j2] -= v[i][j2] * mul;
}
// 注目位置を右下に移す．
i++; j++;
}
// 最後に見つかったピボットの位置が第 m 列ならば解なし．
if (!pivots.empty() && pivots.back() == m) return vector<T>();
// A x = b の特殊解 x0 の構成（任意定数は全て 0 にする）
vector<T> x0(m);
int rnk = sz(pivots);
rep(i, rnk) x0[pivots[i]] = v[i][m];
// 同次形 A x = 0 の一般解 {x} の基底の構成（任意定数を 1-hot にする）
if (xs != nullptr) {
xs->clear();
int i = 0;
rep(j, m) {
if (i < rnk && j == pivots[i]) {
i++;
continue;
}
vector<T> x(m);
x[j] = T(1);
rep(i2, i) x[pivots[i2]] = -v[i2][j];
xs->emplace_back(move(x));
}
}
return x0;
}
// https://qiita.com/satoshin_astonish/items/a628ec64f29e77501d07
namespace satoshin {
/* 内積 */
double dot(const vl& x, const vd& y) {
double z = 0.0;
const int n = sz(x);
for (int i = 0; i < n; ++i) z += x[i] * y[i];
return z;
}
double dot(const vd& x, const vd& y) {
double z = 0.0;
const int n = sz(x);
for (int i = 0; i < n; ++i) z += x[i] * y[i];
return z;
}
double dot(const vl& x, const vl& y) {
double z = 0.0;
const int n = sz(x);
for (int i = 0; i < n; ++i) z += x[i] * y[i];
return z;
}
/* Gram-Schmidtの直交化 */
tuple<vd, vvd> Gram_Schmidt_squared(const vvl& b) {
const int n = sz(b), m = sz(b[0]); int i, j, k;
vd B(n);
vvd GSOb(n, vd(m)), mu(n, vd(n));
for (i = 0; i < n; ++i) {
mu[i][i] = 1.0;
for (j = 0; j < m; ++j) GSOb[i][j] = (double)b[i][j];
for (j = 0; j < i; ++j) {
mu[i][j] = dot(b[i], GSOb[j]) / dot(GSOb[j], GSOb[j]);
for (k = 0; k < m; ++k) GSOb[i][k] -= mu[i][j] * GSOb[j][k];
}
B[i] = dot(GSOb[i], GSOb[i]);
}
return std::forward_as_tuple(B, mu);
}
/* 部分サイズ基底簡約 */
void SizeReduce(vvl& b, vvd& mu, const int i, const int j) {
ll q;
const int m = sz(b[0]);
if (mu[i][j] > 0.5 || mu[i][j] < -0.5) {
q = (ll)round(mu[i][j]);
for (int k = 0; k < m; ++k) b[i][k] -= q * b[j][k];
for (int k = 0; k <= j; ++k) mu[i][k] -= mu[j][k] * q;
}
}
/* LLL基底簡約 */
void LLLReduce(vvl& b, const float d = 0.99) {
const int n = sz(b), m = sz(b[0]); int j, i, h;
double t, nu, BB, C;
auto [B, mu] = Gram_Schmidt_squared(b);
ll tmp;
for (int k = 1; k < n;) {
h = k - 1;
for (j = h; j > -1; --j) SizeReduce(b, mu, k, j);
//Checks if the lattice basis matrix b satisfies Lovasz condition.
if (k > 0 && B[k] < (d - mu[k][h] * mu[k][h]) * B[h]) {
for (i = 0; i < m; ++i) { tmp = b[h][i]; b[h][i] = b[k][i]; b[k][i] = tmp; }
nu = mu[k][h]; BB = B[k] + nu * nu * B[h]; C = 1.0 / BB;
mu[k][h] = nu * B[h] * C; B[k] *= B[h] * C; B[h] = BB;
for (i = 0; i <= k - 2; ++i) {
t = mu[h][i]; mu[h][i] = mu[k][i]; mu[k][i] = t;
}
for (i = k + 1; i < n; ++i) {
t = mu[i][k]; mu[i][k] = mu[i][h] - nu * t;
mu[i][h] = t + mu[k][h] * mu[i][k];
}
--k;
}
else ++k;
}
}
}
vl LLLReduce(const vvm& lat_) {
int h = sz(lat_);
int w = sz(lat_[0]);
vvl lat(h + w, vl(w));
rep(i, h) rep(j, w) lat[i][j] = lat_[i][j].val();
rep(i, w) lat[h + i][i] = mint::mod();
h = sz(lat);
satoshin::LLLReduce(lat);
// L1 ノルムをチェックする．
ll sum = 0;
rep(j, w) sum += abs(lat[0][j]);
dump("L1:", sum);
// L1 ノルムが大きいものは捨てる．
repi(i, 1, h - 1) {
ll sum2 = 0;
rep(j, w) sum2 += abs(lat[i][j]);
if (sum2 > sum * 10.) {
lat.resize(i);
h = i;
break;
}
}
dump("lat:"); frac_print = 1; dumpel(lat); frac_print = 0;
return lat[0];
}
vl LLLReduce2(const vvm& xs) {
int h = sz(xs);
int w = sz(xs[0]);
vl lat0(w);
#ifdef _MSC_VER
string cmd;
cmd += "wolframscript -code \"MOD=";
cmd += to_string(mint::mod());
cmd += ";";
cmd += "SortBy[LatticeReduce@Join[{";
rep(i, h) {
cmd += "{";
rep(j, w) {
cmd += to_string(xs[i][j].val());
cmd += ",";
}
if (cmd.back() == ',') cmd.pop_back();
cmd += "},";
}
if (cmd.back() == ',') cmd.pop_back();
cmd += "},MOD IdentityMatrix[";
cmd += to_string(w);
cmd += "]],N@Norm@# &]\"";
//dump("cmd:", cmd);
FILE* fp = _popen(cmd.c_str(), "r");
char buf[1 << 20];
//while (fgets(buf, sizeof(buf), fp)) printf("%s", buf);
while (fgets(buf, sizeof(buf), fp))
_pclose(fp);
stringstream ss{ buf + 2 };
rep(j, w) {
string s;
getline(ss, s, ' ');
lat0[j] = stol(s);
}
#endif
return lat0;
}
string to_signed_string(mint x) {
int v = x.val();
if (v > mint::mod() / 2) v -= mint::mod();
return to_string(v);
}
pair<vm, int> slice_1D(const vvm& tbl, int i2) {
// seq : tbl[..][i2] を抜き出した列
vm seq;
// offset : seq[0] が tbl[offset][i2] に対応することを表す．
int offset = INF;
rep(i1, sz(tbl)) {
if (i2 < sz(tbl[i1])) {
chmin(offset, i1);
seq.push_back(tbl[i1][i2]);
}
}
return { seq, offset };
}
tuple<vvm, int, int> slice_2D(const vvvm& box, int i3) {
// tbl : box[..][..][i3] を抜き出したテーブル
vvm tbl;
// offset : tbl[0][0] が box[offset1][offset2][i3] に対応することを表す．
int offset1 = INF, offset2 = INF;
rep(i1, sz(box)) {
tbl.push_back(vm());
rep(i2, sz(box[i1])) {
if (i2 < sz(box[i1]) && i3 < sz(box[i1][i2])) {
chmin(offset1, i1);
chmin(offset2, i2);
tbl.back().push_back(box[i1][i2][i3]);
}
}
if (tbl.back().empty()) tbl.pop_back();
}
return { tbl, offset1, offset2 };
}
// 変数係数線形漸化式の係数を計算し，埋め込み用のコードを出力する．
vvm embed_coefs_1D(const vm& seq, int TRM_ini, int DEG_ini, int LLL) {
int n = sz(seq);
// TRM 項間の，係数多項式の次数 DEG 未満の変数係数線形漸化式
// Σt∈[0..TRM) Σd∈[0..DEG) coefs[t][d] (i-TRM+1+t)^d seq[i-t] = 0
// を探す．
int TRM = TRM_ini, DEG = DEG_ini;
int P_MAX = max(TRM, DEG);
while (1) {
//dump("TRM:", TRM, "DEG:", DEG);
int h = n - TRM + 1;
int w = TRM * DEG;
// 行列方程式 A x = 0 を解いて一般解の基底 xs を求める．
Matrix<mint> A(h, w);
repi(i, TRM - 1, n - 1) {
rep(t, TRM) rep(d, DEG) {
A[i - TRM + 1][t * DEG + d] = mint(i - TRM + 1 + t).pow(d) * seq[i - t];
}
}
vvm xs;
gauss_jordan_elimination(A, vm(h), &xs);
// 自明解 x = 0 しか存在しない場合は失敗．
if (xs.empty()) {
while (1) {
DEG++;
if (DEG > P_MAX) { DEG = 1; TRM++; };
if (TRM > P_MAX) { TRM = 1; P_MAX++; };
if (max(TRM, DEG) == P_MAX) break;
}
continue;
}
dump("TRM:", TRM, "DEG:", DEG);
dump("#eq:", h, "#var:", w);
dump("xs:"); frac_print = 1; dumpel(xs); frac_print = 0;
// 変数係数線形漸化式の係数
vvm coefs(TRM, vm(DEG));
if (LLL == 0) {
rep(t, TRM) rep(d, DEG) coefs[t][d] = xs.back()[t * DEG + d];
}
else if (LLL == 1) {
// A x = 0 の解空間の基底に LLL を適用する．
auto lat0 = LLLReduce(xs);
rep(t, TRM) rep(d, DEG) coefs[t][d] = lat0[t * DEG + d];
}
else if (LLL == 2) {
// A x = 0 の解空間の基底に本気の LLL を適用する（埋め込み専用）
auto lat0 = LLLReduce2(xs);
rep(t, TRM) rep(d, DEG) coefs[t][d] = lat0[t * DEG + d];
}
// 分母チェック
#ifdef _MSC_VER
cout << "dnm 1D:" << endl;
string cmd;
cmd += "wolframscript -code \"MOD=";
cmd += to_string(mint::mod());
cmd += ";";
cmd += "toFrac[x_]:=Module[{},Do[num=Mod[x*dnm,MOD,-MOD/2];If[Abs[num]<=Sqrt@MOD,Return[num/dnm,Module]],{dnm,1,Sqrt@MOD}]];";
cmd += "Factor[";
rep(d, DEG) {
cmd += to_string(coefs[0][d].val());
cmd += "*(i1-";
cmd += to_string(TRM);
cmd += "+1)^";
cmd += to_string(d);
cmd += "+";
}
cmd.pop_back();
cmd += ",Modulus->MOD]/.x_Integer:>toFrac[x]\"";
//dump("cmd:", cmd);
FILE* fp = _popen(cmd.c_str(), "r");
char buf[1 << 16];
while (fgets(buf, sizeof(buf), fp)) printf("%s", buf);
_pclose(fp);
#endif
return coefs;
}
return vvm();
}
// 変数係数線形漸化式の係数を計算し，埋め込み用のコードを出力する．
pair<vvvm, vvvm> embed_coefs_2D(const vvm& tbl, int DEG1_ini, int TRM2_ini, int DEG2_ini, int LLL) {
int n1 = sz(tbl);
// TRM 項間の，係数多項式の次数 DEG 未満の変数係数線形漸化式
// Σt1∈[0..TRM1) Σd1∈[0..DEG1) Σt2∈[0..TRM2) Σd2∈[0..DEG2)
// c[t1][d1][t2][d2] (i1-TRM1+1+t1)^d1 (i2-TRM2+1+t2)^d2 tbl[i1-t1][i2-t2] = 0
// を探す．
int TRM1 = 1, DEG1 = DEG1_ini;
int TRM2 = TRM2_ini, DEG2 = DEG2_ini;
int P_MAX = max({ TRM1, DEG1, TRM2, DEG2 });
while (1) {
//dump("TRM1:", TRM1, "DEG1:", DEG1, "TRM2:", TRM2, "DEG2:", DEG2);
int w = TRM1 * DEG1 * TRM2 * DEG2;
// 行列方程式 A x = 0 を解いて一般解の基底 xs を求める．
Matrix<mint> A(0, w);
repi(i1, TRM1 - 1, n1 - 1) {
int n2 = sz(tbl[i1]);
repi(i2, TRM2 - 1, n2 - 1) {
vm a(w); bool valid = true;
rep(t1, TRM1) rep(d1, DEG1) rep(t2, TRM2) rep(d2, DEG2) {
if (i2 - t2 >= sz(tbl[i1 - t1])) {
valid = false;
t1 = TRM1;
d1 = DEG1;
t2 = TRM2;
d2 = DEG2;
break;
}
int idx = ((t1 * DEG1 + d1) * TRM2 + t2) * DEG2 + d2;
mint pow_i = mint(i1 - TRM1 + 1 + t1).pow(d1) * mint(i2 - TRM2 + 1 + t2).pow(d2);
a[idx] = pow_i * tbl[i1 - t1][i2 - t2];
}
if (valid) A.push_back(a);
}
}
int h = A.n;
vvm xs;
gauss_jordan_elimination(A, vm(h), &xs);
// 自明解 x = 0 しか存在しない場合は失敗．
if (xs.empty()) {
while (1) {
DEG2++;
if (DEG2 > P_MAX) { DEG2 = 1; TRM2++; };
if (TRM2 > P_MAX) { TRM2 = 1; DEG1++; };
if (DEG1 > P_MAX) { DEG1 = 1; TRM1++; };
if (TRM1 > 1) { TRM1 = 1; P_MAX++; };
if (max({ TRM1, DEG1, TRM2, DEG2 }) == P_MAX) break;
}
continue;
}
dump("TRM1:", TRM1, "DEG1:", DEG1, "TRM2:", TRM2, "DEG2:", DEG2);
dump("#eq:", h, "#var:", w);
dump("xs:"); frac_print = 1; dumpel(xs); frac_print = 0;
// 変数係数線形漸化式の係数
vvvm coefs(DEG1, vvm(TRM2, vm(DEG2)));
if (LLL == 0) {
rep(d1, DEG1) rep(t2, TRM2) rep(d2, DEG2) {
int idx = (d1 * TRM2 + t2) * DEG2 + d2;
coefs[d1][t2][d2] = xs.back()[idx];
}
}
else if (LLL == 1) {
// A x = 0 の解空間の基底に LLL を適用する．
auto lat0 = LLLReduce(xs);
rep(d1, DEG1) rep(t2, TRM2) rep(d2, DEG2) {
int idx = (d1 * TRM2 + t2) * DEG2 + d2;
coefs[d1][t2][d2] = lat0[idx];
}
}
else if (LLL == 2) {
// A x = 0 の解空間の基底に本気の LLL を適用する（埋め込み専用）
auto lat0 = LLLReduce2(xs);
rep(d1, DEG1) rep(t2, TRM2) rep(d2, DEG2) {
int idx = (d1 * TRM2 + t2) * DEG2 + d2;
coefs[d1][t2][d2] = lat0[idx];
}
}
// 分母チェック
#ifdef _MSC_VER
cout << "dnm 2D:" << endl;
string cmd;
cmd += "wolframscript -code \"MOD=";
cmd += to_string(mint::mod());
cmd += ";";
cmd += "toFrac[x_]:=Module[{},Do[num=Mod[x*dnm,MOD,-MOD/2];If[Abs[num]<=Sqrt@MOD,Return[num/dnm,Module]],{dnm,1,Sqrt@MOD}]];";
cmd += "Factor[";
rep(d2, DEG2) {
rep(d1, DEG1) {
cmd += to_signed_string(coefs[d1][0][d2]);
cmd += "*(i1-";
cmd += to_string(TRM1);
cmd += "+1)^";
cmd += to_string(d1);
cmd += "*(i2-";
cmd += to_string(TRM2);
cmd += "+1)^";
cmd += to_string(d2);
cmd += "+";
}
}
cmd.pop_back();
cmd += "]/.x_Integer:>toFrac[x]\"";
//dump("cmd:", cmd);
FILE* fp = _popen(cmd.c_str(), "r");
char buf[1 << 16];
while (fgets(buf, sizeof(buf), fp)) printf("%s", buf);
_pclose(fp);
#endif
// i1 方向への初項の延長
dump("------- embed_coefs_1D -------");
vvvm coefs1(TRM2 - 1);
rep(i2, TRM2 - 1) {
dump("--- i2:", i2, "---");
auto [seq, offset] = slice_1D(tbl, i2);
coefs1[i2] = embed_coefs_1D(seq, 1, 1, LLL);
}
return { coefs1, coefs };
}
return pair<vvvm, vvvm>();
}
// 変数係数線形漸化式の係数を計算し，埋め込み用のコードを出力する．
tuple<vvvvm, vvvvm, vvvvm> embed_coefs_3D(const vvvm& box, int DEG1_ini, int DEG2_ini, int TRM3_ini, int DEG3_ini, int LLL) {
int n1 = sz(box);
// TRM 項間の，係数多項式の次数 DEG 未満の変数係数線形漸化式
// Σt1∈[0..TRM1) Σd1∈[0..DEG1) Σt2∈[0..TRM2) Σd2∈[0..DEG2) Σt3∈[0..TRM3) Σd3∈[0..DEG3)
// c[t1][d1][t2][d2]][t3][d3] (i1-TRM1+1+t1)^d1 (i2-TRM2+1+t2)^d2 (i3-TRM3+1+t3)^d3 box[i1-t1][i2-t2][i3-t3] = 0
// を探す．
int TRM1 = 1, DEG1 = DEG1_ini;
int TRM2 = 1, DEG2 = DEG2_ini;
int TRM3 = TRM3_ini, DEG3 = DEG3_ini;
int P_MAX = max({ TRM1, DEG1, TRM2, DEG2, TRM3, DEG3 });
while (1) {
//dump("TRM1:", TRM1, "DEG1:", DEG1, "TRM2:", TRM2, "DEG2:", DEG2, "TRM3:", TRM3, "DEG3:", DEG3);
int w = TRM1 * DEG1 * TRM2 * DEG2 * TRM3 * DEG3;
// 行列方程式 A x = 0 を解いて一般解の基底 xs を求める．
Matrix<mint> A(0, w);
repi(i1, TRM1 - 1, n1 - 1) {
int n2 = sz(box[i1]);
repi(i2, TRM2 - 1, n2 - 1) {
int n3 = sz(box[i1][i2]);
repi(i3, TRM3 - 1, n3 - 1) {
vm a(w); bool valid = true;
rep(t1, TRM1) rep(d1, DEG1) rep(t2, TRM2) rep(d2, DEG2) rep(t3, TRM3) rep(d3, DEG3) {
if (i2 - t2 >= sz(box[i1 - t1]) || i3 - t3 >= sz(box[i1 - t1][i2 - t2])) {
valid = false;
t1 = TRM1;
d1 = DEG1;
t2 = TRM2;
d2 = DEG2;
t3 = TRM3;
d3 = DEG3;
break;
}
int idx = ((((t1 * DEG1 + d1) * TRM2 + t2) * DEG2 + d2) * TRM3 + t3) * DEG3 + d3;
mint pow_i = mint(i1 - TRM1 + 1 + t1).pow(d1) * mint(i2 - TRM2 + 1 + t2).pow(d2) * mint(i3 - TRM3 + 1 + t3).pow(d3);
a[idx] = pow_i * box[i1 - t1][i2 - t2][i3 - t3];
}
if (valid) A.push_back(a);
}
}
}
int h = A.n;
vvm xs;
gauss_jordan_elimination(A, vm(h), &xs);
// 自明解 x = 0 しか存在しない場合は失敗．
if (xs.empty()) {
while (1) {
DEG3++;
if (DEG3 > P_MAX) { DEG3 = 1; TRM3++; };
if (TRM3 > P_MAX) { TRM3 = 1; DEG2++; };
if (DEG2 > P_MAX) { DEG2 = 1; TRM2++; };
if (TRM2 > 1) { TRM2 = 1; DEG1++; };
if (DEG1 > P_MAX) { DEG1 = 1; TRM1++; };
if (TRM1 > 1) { TRM1 = 1; P_MAX++; };
if (max({ TRM1, DEG1, TRM2, DEG2, TRM3, DEG3 }) == P_MAX) break;
}
continue;
}
dump("TRM1:", TRM1, "DEG1:", DEG1, "TRM2:", TRM2, "DEG2:", DEG2, "TRM3:", TRM3, "DEG3:", DEG3);
dump("#eq:", h, "#var:", w);
dump("xs:"); frac_print = 1; dumpel(xs); frac_print = 0;
// 変数係数線形漸化式の係数
vvvvm coefs3(DEG1, vvvm(DEG2, vvm(TRM3, vm(DEG3))));
if (LLL == 0) {
rep(d1, DEG1) rep(d2, DEG2) rep(t3, TRM3) rep(d3, DEG3) {
int idx = ((d1 * DEG2 + d2) * TRM3 + t3) * DEG3 + d3;
coefs3[d1][d2][t3][d3] = xs.back()[idx];
}
}
else if (LLL == 1) {
// A x = 0 の解空間の基底に LLL を適用する．
auto lat0 = LLLReduce(xs);
rep(d1, DEG1) rep(d2, DEG2) rep(t3, TRM3) rep(d3, DEG3) {
int idx = ((d1 * DEG2 + d2) * TRM3 + t3) * DEG3 + d3;
coefs3[d1][d2][t3][d3] = lat0[idx];
}
}
else if (LLL == 2) {
// A x = 0 の解空間の基底に本気の LLL を適用する（埋め込み専用）
auto lat0 = LLLReduce2(xs);
rep(d1, DEG1) rep(d2, DEG2) rep(t3, TRM3) rep(d3, DEG3) {
int idx = ((d1 * DEG2 + d2) * TRM3 + t3) * DEG3 + d3;
coefs3[d1][d2][t3][d3] = lat0[idx];
}
}
// 分母チェック
#ifdef _MSC_VER
cout << "dnm 3D:" << endl;
string cmd;
cmd += "wolframscript -code \"MOD=";
cmd += to_string(mint::mod());
cmd += ";";
cmd += "toFrac[x_]:=Module[{},Do[num=Mod[x*dnm,MOD,-MOD/2];If[Abs[num]<=Sqrt@MOD,Return[num/dnm,Module]],{dnm,1,Sqrt@MOD}]];";
cmd += "Factor[";
rep(d3, DEG3) {
rep(d2, DEG2) {
rep(d1, DEG1) {
cmd += to_signed_string(coefs3[d1][d2][0][d3]);
cmd += "*(i1-";
cmd += to_string(TRM1);
cmd += "+1)^";
cmd += to_string(d1);
cmd += "*(i2-";
cmd += to_string(TRM2);
cmd += "+1)^";
cmd += to_string(d2);
cmd += "*(i3-";
cmd += to_string(TRM3);
cmd += "+1)^";
cmd += to_string(d3);
cmd += "+";
}
}
}
cmd.pop_back();
cmd += "]/.x_Integer:>toFrac[x]\"";
//dump("cmd:", cmd);
FILE* fp = _popen(cmd.c_str(), "r");
char buf[4096];
while (fgets(buf, sizeof(buf), fp)) printf("%s", buf);
_pclose(fp);
#endif
// (i1,i2) 平面における初項の延長
dump("------- embed_coefs_2D -------");
vvvvm coefs1(TRM3 - 1); vvvvm coefs2(TRM3 - 1);
rep(i3, TRM3 - 1) {
dump("--- i3:", i3, "---");
auto [tbl, offset1, offset2] = slice_2D(box, i3);
tie(coefs1[i3], coefs2[i3]) = embed_coefs_2D(tbl, 1, 1, 1, LLL);
}
#ifdef _MSC_VER
// 埋め込み用の文字列を出力する．
string eb;
eb += "\n";
eb += "constexpr int TRM1 = ";
eb += to_string(TRM1);
eb += ";\n";
eb += "constexpr int DEG1 = ";
eb += to_string(DEG1);
eb += ";\n";
eb += "constexpr int TRM2 = ";
eb += to_string(TRM2);
eb += ";\n";
eb += "constexpr int DEG2 = ";
eb += to_string(DEG2);
eb += ";\n";
eb += "constexpr int TRM3 = ";
eb += to_string(TRM3);
eb += ";\n";
eb += "constexpr int DEG3 = ";
eb += to_string(DEG3);
eb += ";\n\n";
eb += "vvvm coefs1[TRM3 - 1] = {";
rep(i3, TRM3 - 1) {
eb += "{\n";
rep(i2, sz(coefs1[i3])) {
eb += "{";
rep(t1, sz(coefs1[i3][i2])) {
eb += "{";
rep(d1, sz(coefs1[i3][i2][t1])) {
eb += to_signed_string(coefs1[i3][i2][t1][d1]) + ",";
}
eb.pop_back();
eb += "},";
}
eb.pop_back();
eb += "},\n";
}
eb.pop_back(); eb.pop_back();
eb += "},";
}
eb.pop_back();
eb += "};\n\n";
eb += "vvvm coefs2[TRM3 - 1] = {";
rep(i3, TRM3 - 1) {
eb += "{\n";
rep(d1, sz(coefs2[i3])) {
eb += "{";
rep(t2, sz(coefs2[i3][d1])) {
eb += "{";
rep(d2, sz(coefs2[i3][d1][t2])) {
eb += to_signed_string(coefs2[i3][d1][t2][d2]) + ",";
}
eb.pop_back();
eb += "},";
}
eb.pop_back();
eb += "},\n";
}
eb.pop_back(); eb.pop_back();
eb += "},";
}
eb.pop_back();
eb += "};\n\n";
eb += "mint coefs3[DEG1][DEG2][TRM3][DEG3] = {";
rep(d1, DEG1) {
eb += "{\n";
rep(d2, DEG2) {
eb += "{";
rep(t3, TRM3) {
eb += "{";
rep(d3, DEG3) {
eb += to_signed_string(coefs3[d1][d2][t3][d3]) + ",";
}
eb.pop_back();
eb += "},";
}
eb.pop_back();
eb += "},\n";
}
eb.pop_back(); eb.pop_back();
eb += "},";
}
eb.pop_back();
eb += "};\n\n";
cout << eb;
#endif
return { coefs1, coefs2, coefs3 };
}
return tuple<vvvvm, vvvvm, vvvvm>();
}
// 数列 seq を延長して seq[0..N] にする．
void solve_1D(vm& seq, int N, const vvm& coefs) {
// static int call_cnt = 0;
int TRM = sz(coefs);
int DEG = sz(coefs[0]);
int n = sz(seq);
seq.resize(N + 1);
// TRM 項間の，係数多項式の次数 DEG 未満の変数係数線形漸化式
// Σt∈[0..TRM) Σd∈[0..DEG) coefs[t][f] (i-TRM+1+t)^d a[i-t] = 0
// を用いて数列 a を延長する．
repi(i, n, N) {
mint dnm = 0;
mint pow_i = 1;
rep(d, DEG) {
dnm += coefs[0][d] * pow_i;
pow_i *= i - TRM + 1;
}
mint num = 0;
repi(t, 1, TRM - 1) {
mint pow_i = 1;
rep(d, DEG) {
num += coefs[t][d] * pow_i * seq[i - t];
pow_i *= i - TRM + 1 + t;
}
}
// dnm * a[i] + num = 0 を解く．分母 0 に注意！
if (dnm == 0) {
dump("DIVISION BY ZERO at i1 =", i);
Assert(dnm != 0);
}
seq[i] = -num / dnm;
// 除算回避用
//mint dnm_inv;
//if (call_cnt == 0) {
// dnm_inv = fm.inv_neg(1 + i);
//}
//else {
// dnm_inv = fm.inv_neg(-1 + 2 * i) * 2 * fm.inv_neg(2 + i);
//}
//seq[i] = -num * dnm_inv;
}
// call_cnt++;
}
// 2 次元数列 tbl を元に seq = tbl[N][0..M] を計算する．
vm solve_2D(const vvm& tbl, int N, int M, const vvvm& coefs1, const vvvm& coefs) {
// static int call_cnt = 0;
int TRM1 = 1;
int DEG1 = sz(coefs);
int TRM2 = sz(coefs[0]);
int DEG2 = sz(coefs[0][0]);
vm res(TRM2 - 1);
// i1 方向に tbl[..][0], ..., tbl[..][TRM2-2] を延長する．
dump("------- solve_1D -------");
rep(i2, TRM2 - 1) {
dump("--- i2:", i2, "---");
auto [seq, offset] = slice_1D(tbl, i2);
if (N - offset < 0) continue;
solve_1D(seq, N - offset, coefs1[i2]);
res[i2] = seq[N - offset];
}
vm pow_i1s(DEG1);
pow_i1s[0] = 1;
repi(d1, 1, DEG1 - 1) pow_i1s[d1] = pow_i1s[d1 - 1] * (N - TRM1 + 1);
// i2 方向に tbl[N][..] を延長する．
res.resize(M + 1);
repi(i2, TRM2 - 1, M) {
mint dnm = 0;
mint pow_i2 = 1;
rep(d2, DEG2) {
rep(d1, DEG1) {
dnm += coefs[d1][0][d2] * pow_i1s[d1] * pow_i2;
}
pow_i2 *= i2 - TRM2 + 1;
}
mint num = 0;
repi(t2, 1, TRM2 - 1) {
mint pow_i2 = 1;
rep(d2, DEG2) {
rep(d1, DEG1) {
num += coefs[d1][t2][d2] * pow_i1s[d1] * pow_i2 * res[i2 - t2];
}
pow_i2 *= i2 - TRM2 + 1 + t2;
}
}
// dnm * tbl[N][i2] + num = 0 を解く．分母 0 に注意！
if (dnm == 0) {
dump("DIVISION BY ZERO at N1 =", N, "i2 =", i2);
Assert(dnm != 0);
}
res[i2] = -num / dnm;
}
// call_cnt++;
return res;
}
// 3 次元数列 box を元に seq = tbl[N1][N2][0..N3] を計算する．
vm solve_3D(const vvvm& box, int N1, int N2, int N3, const vvvvm& coefs1, const vvvvm& coefs2, const vvvvm& coefs3) {
int TRM1 = 1;
int DEG1 = sz(coefs3);
int TRM2 = 1;
int DEG2 = sz(coefs3[0]);
int TRM3 = sz(coefs3[0][0]);
int DEG3 = sz(coefs3[0][0][0]);
vm res(TRM3 - 1);
// (i1,i2) 平面における延長
dump("------- solve_2D -------");
rep(i3, TRM3 - 1) {
dump("--- i3:", i3, "---");
auto [tbl, offset1, offset2] = slice_2D(box, i3);
if (N1 - offset1 < 0 || N2 - offset2 < 0) continue;
auto seq = solve_2D(tbl, N1 - offset1, N2 - offset2, coefs1[i3], coefs2[i3]);
res[i3] = seq[N2 - offset2];
}
vm pow_i1s(DEG1);
pow_i1s[0] = 1;
repi(d1, 1, DEG1 - 1) pow_i1s[d1] = pow_i1s[d1 - 1] * (N1 - TRM1 + 1);
vm pow_i2s(DEG2);
pow_i2s[0] = 1;
repi(d2, 1, DEG2 - 1) pow_i2s[d2] = pow_i2s[d2 - 1] * (N2 - TRM2 + 1);
// i3 方向に tbl[N1][N2][..] を延長する．
res.resize(N3 + 1);
repi(i3, TRM3 - 1, N3) {
mint dnm = 0;
mint pow_i3 = 1;
rep(d3, DEG3) {
rep(d2, DEG2) {
rep(d1, DEG1) {
dnm += coefs3[d1][d2][0][d3] * pow_i1s[d1] * pow_i2s[d2] * pow_i3;
}
}
pow_i3 *= i3 - TRM3 + 1;
}
mint num = 0;
repi(t3, 1, TRM3 - 1) {
mint pow_i3 = 1;
rep(d3, DEG3) {
rep(d2, DEG2) {
rep(d1, DEG1) {
num += coefs3[d1][d2][t3][d3] * pow_i1s[d1] * pow_i2s[d2] * pow_i3 * res[i3 - t3];
}
}
pow_i3 *= i3 - TRM3 + 1 + t3;
}
}
// dnm * tbl[N1][N2][i3] + num = 0 を解く．分母 0 に注意！
if (dnm == 0) {
dump("DIVISION BY ZERO at N1 =", N1, "N2 =", N2, "i3 =", i3);
Assert(dnm != 0);
}
res[i3] = -num / dnm;
}
return res;
}
// 3 次元数列 box を元に seq = tbl[N1][N2][0..N3] を計算する．
vm solve_3D(const vvvm& box, int N1, int N2, int N3) {
// --------------- embed_coefs() からの出力を貼る ----------------
constexpr int TRM1 = 1;
constexpr int DEG1 = 2;
constexpr int TRM2 = 1;
constexpr int DEG2 = 2;
constexpr int TRM3 = 2;
constexpr int DEG3 = 2;
vvvm coefs1[TRM3 - 1] = { {
{{-1},{1}}} };
vvvm coefs2[TRM3 - 1] = { {
{{-1},{1}}} };
mint coefs3[DEG1][DEG2][TRM3][DEG3] = { {
{{-1,0},{1,0}},
{{0,0},{0,0}}},{
{{0,0},{0,0}},
{{0,-1},{0,1}}} };
// --------------------------------------------------------------
vm res(TRM3 - 1);
// (i1,i2) 平面における延長
dump("------- solve_2D -------");
rep(i3, TRM3 - 1) {
dump("--- i3:", i3, "---");
auto [tbl, offset1, offset2] = slice_2D(box, i3);
if (N1 - offset1 < 0 || N2 - offset2 < 0) continue;
auto seq = solve_2D(tbl, N1 - offset1, N2 - offset2, coefs1[i3], coefs2[i3]);
res[i3] = seq[N2 - offset2];
}
vm pow_i1s(DEG1);
pow_i1s[0] = 1;
repi(d1, 1, DEG1 - 1) pow_i1s[d1] = pow_i1s[d1 - 1] * (N1 - TRM1 + 1);
vm pow_i2s(DEG2);
pow_i2s[0] = 1;
repi(d2, 1, DEG2 - 1) pow_i2s[d2] = pow_i2s[d2 - 1] * (N2 - TRM2 + 1);
// i3 方向に tbl[N1][N2][..] を延長する．
res.resize(N3 + 1);
repi(i3, TRM3 - 1, N3) {
mint dnm = 0;
mint pow_i3 = 1;
rep(d3, DEG3) {
rep(d2, DEG2) {
rep(d1, DEG1) {
dnm += coefs3[d1][d2][0][d3] * pow_i1s[d1] * pow_i2s[d2] * pow_i3;
}
}
pow_i3 *= i3 - TRM3 + 1;
}
mint num = 0;
repi(t3, 1, TRM3 - 1) {
mint pow_i3 = 1;
rep(d3, DEG3) {
rep(d2, DEG2) {
rep(d1, DEG1) {
num += coefs3[d1][d2][t3][d3] * pow_i1s[d1] * pow_i2s[d2] * pow_i3 * res[i3 - t3];
}
}
pow_i3 *= i3 - TRM3 + 1 + t3;
}
}
// dnm * tbl[N1][N2][i3] + num = 0 を解く．分母 0 に注意！
if (dnm == 0) {
dump("DIVISION BY ZERO at N1 =", N1, "N2 =", N2, "i3 =", i3);
Assert(dnm != 0);
}
res[i3] = -num / dnm;
}
return res;
}
int main() {
// input_from_file("input.txt");
// output_to_file("output.txt");
//【方法】
// 愚直を書いて集めたデータをもとに変数係数線形漸化式を復元する．
//【使い方】
// 1. vvm tbl = naive() を実装する．
// 2. embed_coefs() を実行する．
// 3. 出力を solve() 内に貼る．
// 4. solve(tbl, n, m) で勝手に tbl[n][0..m] を求めてくれる．
// 愚直解を用意する．再計算がイヤなら埋め込む．
auto box = naive();
// 愚直解を渡して変数係数線形漸化式の係数を得る．再計算がイヤなら埋め込む．
// 引数：tbl, DEG1_ini, DEG2_ini, TRM3_ini, DEG3_ini, LLL
auto [coefs1, coefs2, coefs3] = embed_coefs_3D(box, 1, 1, 1, 1, 0);
int n, a1, a2;
cin >> n >> a1 >> a2;
if (a1 == a2) {
mint res = n;
repi(i, 1, n) res *= i;
EXIT(res);
}
if (a1 > a2) swap(a1, a2);
int n0 = a1 - 1, n1 = a2 - a1 - 1, n2 = n - a2 - 1;
// 3 次元数列 box を元に seq = box[n1][n2][0..n3] を計算する．
// 整理すると綺麗な式になるなら FullSimplify[] すると速くなる．
auto seq = solve_3D(box, n0, n1, n2, coefs1, coefs2, coefs3);
// auto seq = solve_3D(box, n0, n1, n2);
// dump(seq);
mint res = seq[n2];
EXIT(res);
}