Problem 2. Exercise

USACO 2020 US Open Contest, Platinum

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 
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);
		ull r = a - q * b; // can be proven that 0 <= 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
}

INPUT FORMAT (file exercise.in):

The first line contains $N$ and $M$.

OUTPUT FORMAT (file exercise.out):

A single integer.

SAMPLE INPUT:

5 1000000007

SAMPLE OUTPUT:

369329541

For each $1\le i\le N$, the $i$-th element of the following array is the number of permutations that cause the cows to take $i$ steps: $[1,25,20,30,24,20].$ The answer is $1^1\cdot 2^{25}\cdot 3^{20}\cdot 4^{30}\cdot 5^{24}\cdot 6^{20}\equiv 369329541\pmod{10^9+7}$.

Note: This problem has an expanded memory limit of 512 MB.

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