E2. Abnormal Permutation Pairs (hard version)
time limit per test
4 seconds
memory limit per test
512 megabytes
standard input
standard output

This is the hard version of the problem. The only difference between the easy version and the hard version is the constraints on $$$n$$$. You can only make hacks if both versions are solved.

A permutation of $$$1, 2, \ldots, n$$$ is a sequence of $$$n$$$ integers, where each integer from $$$1$$$ to $$$n$$$ appears exactly once. For example, $$$[2,3,1,4]$$$ is a permutation of $$$1, 2, 3, 4$$$, but $$$[1,4,2,2]$$$ isn't because $$$2$$$ appears twice in it.

Recall that the number of inversions in a permutation $$$a_1, a_2, \ldots, a_n$$$ is the number of pairs of indices $$$(i, j)$$$ such that $$$i < j$$$ and $$$a_i > a_j$$$.

Let $$$p$$$ and $$$q$$$ be two permutations of $$$1, 2, \ldots, n$$$. Find the number of permutation pairs $$$(p,q)$$$ that satisfy the following conditions:

  • $$$p$$$ is lexicographically smaller than $$$q$$$.
  • the number of inversions in $$$p$$$ is greater than the number of inversions in $$$q$$$.

Print the number of such pairs modulo $$$mod$$$. Note that $$$mod$$$ may not be a prime.


The only line contains two integers $$$n$$$ and $$$mod$$$ ($$$1\le n\le 500$$$, $$$1\le mod\le 10^9$$$).


Print one integer, which is the answer modulo $$$mod$$$.

4 403458273

The following are all valid pairs $$$(p,q)$$$ when $$$n=4$$$.

  • $$$p=[1,3,4,2]$$$, $$$q=[2,1,3,4]$$$,
  • $$$p=[1,4,2,3]$$$, $$$q=[2,1,3,4]$$$,
  • $$$p=[1,4,3,2]$$$, $$$q=[2,1,3,4]$$$,
  • $$$p=[1,4,3,2]$$$, $$$q=[2,1,4,3]$$$,
  • $$$p=[1,4,3,2]$$$, $$$q=[2,3,1,4]$$$,
  • $$$p=[1,4,3,2]$$$, $$$q=[3,1,2,4]$$$,
  • $$$p=[2,3,4,1]$$$, $$$q=[3,1,2,4]$$$,
  • $$$p=[2,4,1,3]$$$, $$$q=[3,1,2,4]$$$,
  • $$$p=[2,4,3,1]$$$, $$$q=[3,1,2,4]$$$,
  • $$$p=[2,4,3,1]$$$, $$$q=[3,1,4,2]$$$,
  • $$$p=[2,4,3,1]$$$, $$$q=[3,2,1,4]$$$,
  • $$$p=[2,4,3,1]$$$, $$$q=[4,1,2,3]$$$,
  • $$$p=[3,2,4,1]$$$, $$$q=[4,1,2,3]$$$,
  • $$$p=[3,4,1,2]$$$, $$$q=[4,1,2,3]$$$,
  • $$$p=[3,4,2,1]$$$, $$$q=[4,1,2,3]$$$,
  • $$$p=[3,4,2,1]$$$, $$$q=[4,1,3,2]$$$,
  • $$$p=[3,4,2,1]$$$, $$$q=[4,2,1,3]$$$.