Number of inversions

A permutation is a sequence of $n$ distinct integers from $1$ to $n$. For example, (5, 3, 2, 1, 4) is a permutation. An inversion in a permutation $p$ is an ordered pair of indices $(i,j)$ such that i < j and p_i > p_j. In the example above, the pair of indices $(2,4)$ forms an inversion because 2 < 4 and 3 > 1.

You are given two integers $n$ and $t$. Find the number of permutations of $n$ elements with exactly $t$ inversions. Output the lexicographically smallest permutation with exactly $t$ inversions.