Recall that the permutation is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ($2$ appears twice in the array) and $[1,3,4]$ is also not a permutation ($n=3$ but there is $4$ in the array).
A sequence $a$ is a subsegment of a sequence $b$ if $a$ can be obtained from $b$ by deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end. We will denote the subsegments as $[l, r]$, where $l, r$ are two integers with $1 \le l \le r \le n$. This indicates the subsegment where $l-1$ elements from the beginning and $n-r$ elements from the end are deleted from the sequence.
For a permutation $p_1, p_2, \ldots, p_n$, we define a framed segment as a subsegment $[l,r]$ where $\max\{p_l, p_{l+1}, \dots, p_r\} – \min\{p_l, p_{l+1}, \dots, p_r\} = r – l$. For example, for the permutation $(6, 7, 1, 8, 5, 3, 2, 4)$ some of its framed segments are: $[1, 2], [5, 8], [6, 7], [3, 3], [8, 8]$. In particular, a subsegment $[i,i]$ is always a framed segments for any $i$ between $1$ and $n$, inclusive.
We define the happiness of a permutation $p$ as the number of pairs $(l, r)$ such that $1 \le l \le r \le n$, and $[l, r]$ is a framed segment. For example, the permutation $[3, 1, 2]$ has happiness $5$: all segments except $[1, 2]$ are framed segments.
Given integers $n$ and $m$, Jongwon wants to compute the sum of happiness for all permutations of length $n$, modulo the prime number $m$. Note that there exist $n!$ (factorial of $n$) different permutations of length $n$.Input
The only line contains two integers $n$ and $m$ ($1 \le n \le 250\,000$, $10^8 \le m \le 10^9$, $m$ is prime).Output
Print $r$ ($0 \le r < m$), the sum of happiness for all permutations of length $n$, modulo a prime number $m$.Examplesinput
1 993244853
output
1
input
2 993244853
output
6
input
3 993244853
output
32
input
2019 993244853
output
923958830
input
2020 437122297
output
265955509
Note
For sample input $n=3$, let’s consider all permutations of length $3$:
- $[1, 2, 3]$, all subsegments are framed segment. Happiness is $6$.
- $[1, 3, 2]$, all subsegments except $[1, 2]$ are framed segment. Happiness is $5$.
- $[2, 1, 3]$, all subsegments except $[2, 3]$ are framed segment. Happiness is $5$.
- $[2, 3, 1]$, all subsegments except $[2, 3]$ are framed segment. Happiness is $5$.
- $[3, 1, 2]$, all subsegments except $[1, 2]$ are framed segment. Happiness is $5$.
- $[3, 2, 1]$, all subsegments are framed segment. Happiness is $6$.
Thus, the sum of happiness is $6+5+5+5+5+6 = 32$.
Solution:
#include <bits/stdc++.h> using namespace std; template <typename T> T inverse(T a, T m) { T u = 0, v = 1; while (a != 0) { T t = m / a; m -= t * a; swap(a, m); u -= t * v; swap(u, v); } assert(m == 1); return u; } template <typename T> class Modular { public: using Type = typename decay<decltype(T::value)>::type; constexpr Modular() : value() {} template <typename U> Modular(const U& x) { value = normalize(x); } template <typename U> static Type normalize(const U& x) { Type v; if (-mod() <= x && x < mod()) v = static_cast<Type>(x); else v = static_cast<Type>(x % mod()); if (v < 0) v += mod(); return v; } const Type& operator()() const { return value; } template <typename U> explicit operator U() const { return static_cast<U>(value); } constexpr static Type mod() { return T::value; } Modular& operator+=(const Modular& other) { if ((value += other.value) >= mod()) value -= mod(); return *this; } Modular& operator-=(const Modular& other) { if ((value -= other.value) < 0) value += mod(); return *this; } template <typename U> Modular& operator+=(const U& other) { return *this += Modular(other); } template <typename U> Modular& operator-=(const U& other) { return *this -= Modular(other); } Modular& operator++() { return *this += 1; } Modular& operator--() { return *this -= 1; } Modular operator++(int) { Modular result(*this); *this += 1; return result; } Modular operator--(int) { Modular result(*this); *this -= 1; return result; } Modular operator-() const { return Modular(-value); } template <typename U = T> typename enable_if<is_same<typename Modular<U>::Type, int>::value, Modular>::type& operator*=(const Modular& rhs) { #ifdef _WIN32 uint64_t x = static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value); uint32_t xh = static_cast<uint32_t>(x >> 32), xl = static_cast<uint32_t>(x), d, m; asm( "divl %4; \n\t" : "=a" (d), "=d" (m) : "d" (xh), "a" (xl), "r" (mod()) ); value = m; #else value = normalize(static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value)); #endif return *this; } template <typename U = T> typename enable_if<is_same<typename Modular<U>::Type, int64_t>::value, Modular>::type& operator*=(const Modular& rhs) { int64_t q = static_cast<int64_t>(static_cast<long double>(value) * rhs.value / mod()); value = normalize(value * rhs.value - q * mod()); return *this; } template <typename U = T> typename enable_if<!is_integral<typename Modular<U>::Type>::value, Modular>::type& operator*=(const Modular& rhs) { value = normalize(value * rhs.value); return *this; } Modular& operator/=(const Modular& other) { return *this *= Modular(inverse(other.value, mod())); } template <typename U> friend const Modular<U>& abs(const Modular<U>& v) { return v; } template <typename U> friend bool operator==(const Modular<U>& lhs, const Modular<U>& rhs); template <typename U> friend bool operator<(const Modular<U>& lhs, const Modular<U>& rhs); template <typename U> friend std::istream& operator>>(std::istream& stream, Modular<U>& number); private: Type value; }; template <typename T> bool operator==(const Modular<T>& lhs, const Modular<T>& rhs) { return lhs.value == rhs.value; } template <typename T, typename U> bool operator==(const Modular<T>& lhs, U rhs) { return lhs == Modular<T>(rhs); } template <typename T, typename U> bool operator==(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) == rhs; } template <typename T> bool operator!=(const Modular<T>& lhs, const Modular<T>& rhs) { return !(lhs == rhs); } template <typename T, typename U> bool operator!=(const Modular<T>& lhs, U rhs) { return !(lhs == rhs); } template <typename T, typename U> bool operator!=(U lhs, const Modular<T>& rhs) { return !(lhs == rhs); } template <typename T> bool operator<(const Modular<T>& lhs, const Modular<T>& rhs) { return lhs.value < rhs.value; } template <typename T> Modular<T> operator+(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) += rhs; } template <typename T, typename U> Modular<T> operator+(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) += rhs; } template <typename T, typename U> Modular<T> operator+(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) += rhs; } template <typename T> Modular<T> operator-(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) -= rhs; } template <typename T, typename U> Modular<T> operator-(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) -= rhs; } template <typename T, typename U> Modular<T> operator-(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) -= rhs; } template <typename T> Modular<T> operator*(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) *= rhs; } template <typename T, typename U> Modular<T> operator*(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) *= rhs; } template <typename T, typename U> Modular<T> operator*(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) *= rhs; } template <typename T> Modular<T> operator/(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) /= rhs; } template <typename T, typename U> Modular<T> operator/(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) /= rhs; } template <typename T, typename U> Modular<T> operator/(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) /= rhs; } template<typename T, typename U> Modular<T> power(const Modular<T>& a, const U& b) { assert(b >= 0); Modular<T> x = a, res = 1; U p = b; while (p > 0) { if (p & 1) res *= x; x *= x; p >>= 1; } return res; } template <typename T> bool IsZero(const Modular<T>& number) { return number() == 0; } template <typename T> string to_string(const Modular<T>& number) { return to_string(number()); } template <typename T> std::ostream& operator<<(std::ostream& stream, const Modular<T>& number) { return stream << number(); } template <typename T> std::istream& operator>>(std::istream& stream, Modular<T>& number) { typename common_type<typename Modular<T>::Type, int64_t>::type x; stream >> x; number.value = Modular<T>::normalize(x); return stream; } using ModType = int; struct VarMod { static ModType value; }; ModType VarMod::value; ModType& md = VarMod::value; using Mint = Modular<VarMod>; int main() { ios::sync_with_stdio(false); cin.tie(0); int n; cin >> n >> md; vector<Mint> fact(n + 1); fact[0] = 1; for (int i = 1; i <= n; i++) { fact[i] = fact[i - 1] * i; } Mint ans = 0; for (int k = 1; k <= n; k++) { Mint cur = (n - k + 1) * fact[k] * (n - k + 1) * fact[n - k]; ans += cur; } cout << ans << '\n'; return 0; }