Random permutation graph and Mahonian numbers

Question

Let $K_m$ be the number of m-cliques, $m \in N$, in a random permutation graph $G_n$ with $n$ vertices and $\pi_n$ is the corresponding permutation representation in $S_n$. Let $K_1=n$, and $K_2$ is the number of edges in $G_n$ is the number of inversions in $\pi_n$ (denoted by $Inv(\pi_n)$). In other words the matter of understanding is that a 1−clique is a vertex and there are n of them and the 2−clique is an edge and by construction there are $Inv(π_n)$.

I wonder if we can assosiate the such random permutation graph with Mahonian numbers from OEIS A008302? If yes, how to describe it by math language?

Mahonian numbers $T(n,k)$ presents the number of permutations of {1..n} with k inversions. On the other hand the permutation graph presents $Inv(π_n)$ statistic.

Answer 1

It took me several readings to understand exactly what the question is, but I think that the connection could be described simply as

T(n,k) counts (labelled) permutation graphs with n vertices and k edges.

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However, I'm not sure whether this is sufficiently different from the existing comments on A008302 to be worth adding to that OEIS entry. It's really just

T(n,k) is the number of permutations of {1..n} with k inversions.

or

T(n,k) = number of permutations of (1,2,...,n) having disorder equal to k.

in different words.