E2. Abnormal Permutation Pairs (hard version)
combinatorics; dp; fft; math
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 \(mod\).
The following are all valid pairs \((p,q)\) when \(n=4\). \(p=[1,3,4,2]\), \(q=[2,1,3,4]\), \(p=[1,4,2,3]\), \(q=[2,1,3,4]\), \(p=[1,4,3,2]\), \(q=[2,1,3,4]\), \(p=[1,4,3,2]\), \(q=[2,1,4,3]\), \(p=[1,4,3,2]\), \(q=[2,3,1,4]\), \(p=[1,4,3,2]\), \(q=[3,1,2,4]\), \(p=[2,3,4,1]\), \(q=[3,1,2,4]\), \(p=[2,4,1,3]\), \(q=[3,1,2,4]\), \(p=[2,4,3,1]\), \(q=[3,1,2,4]\), \(p=[2,4,3,1]\), \(q=[3,1,4,2]\), \(p=[2,4,3,1]\), \(q=[3,2,1,4]\), \(p=[2,4,3,1]\), \(q=[4,1,2,3]\), \(p=[3,2,4,1]\), \(q=[4,1,2,3]\), \(p=[3,4,1,2]\), \(q=[4,1,2,3]\), \(p=[3,4,2,1]\), \(q=[4,1,2,3]\), \(p=[3,4,2,1]\), \(q=[4,1,3,2]\), \(p=[3,4,2,1]\), \(q=[4,2,1,3]\).