Exercise timer 2초 memory 512MB

Farmer John has come up with a new morning exercise routine for the cows (again)!

As before, Farmer John's N cows (1\le N\le 7500) are standing in a line. The i-th cow from the left has label i for each 1\le i\le N. He tells them to repeat the following step until the cows are in the same order as when they started.

• Given a permutation A of length N, the cows change their order such that the i-th cow from the left before the change is A_i-th from the left after the change.

For example, if A=(1,2,3,4,5) then the cows perform one step and immediately return to the same order. If A=(2,3,1,5,4), then the cows perform six steps before returning to the original order. The order of the cows from left to right after each step is as follows:

• 0 steps: (1,2,3,4,5)

• 1 step: (3,1,2,5,4)

• 2 steps: (2,3,1,4,5)

• 3 steps: (1,2,3,5,4)

• 4 steps: (3,1,2,4,5)

• 5 steps: (2,3,1,5,4)

• 6 steps: (1,2,3,4,5)

Compute the product of the numbers of steps needed over all N! possible permutations A of length N.

As this number may be very large, output the answer modulo M (10^8\le M\le 10^9+7, M is prime).

Contestants using C++ may find the following code from KACTL helpful. Known as the Barrett reduction, it allows you to compute a \% b several times faster than usual, where b>1 is constant but not known at compile time. (we are not aware of such an optimization for Java, unfortunately).

#include <bits/stdc++.h>
using namespace std;

typedef unsigned long long ull;
typedef __uint128_t L;
struct FastMod {
	ull b, m;
	FastMod(ull b) : b(b), m(ull((L(1) << 64) / b)) {}
	ull reduce(ull a) {
		ull q = (ull)((L(m) * a) >> 64);
		ull r = a - q * b; // can be proven that 0 <= r < 2*b
		return r >= b ? r - b : r;
	}
};
FastMod F(2);

int main() {
	int M = 1000000007; F = FastMod(M);
	ull x = 10ULL*M+3; 
	cout << x << " " << F.reduce(x) << "\n"; // 10000000073 3
}

SCORING:

• Test case 2 satisfies N=8.

• Test cases 3-5 satisfy N\le 50.

• Test cases 6-8 satisfy N\le 500.

• Test cases 9-12 satisfy N\le 3000.

• Test cases 13-16 satisfy no additional constraints.

Problem credits: Benjamin Qi