Let $T(n,k)$ is the number of permutations of numbers $1, ..., n$ and each of the permutations has $k$ inversions. We can consider a table for $T(n,k)$ for some $n$ and $k$. For eg. $n=1,...,6$, $k=1,...,15$:

$1;$

$1, 1;$

$1, 2, 2, 1;$

$1, 3, 5, 6, 5, 3, 1;$

$1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1;$

$1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1;$

If we look at numbers $1, 2, 6, 22, 101$ and so on, we can see the structure like $n-1/2$. I am trying to find other similar structures in the table and very hope that there is a way to get a geometric interpetation. In other words how geometries can help here?

I find the question is intresting due to some physics interpretation. For instance,

1) $n+1/2$ - structure for quantum harmonic oscillator,

2) Triangular numbers: $a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.$ form the number of levels with energy $n+1/2+1=n+3/2$ (in units of h*f0, with Planck's constant h and the oscillator frequency f0) of the three dimensional isotropic harmonic quantum oscillator.

3) The similar family of permutations have arisen in the study of squeezed states of the simple harmonic oscillator. The $n+1/2$ properties is connected to symmetry. It looks like there is a universal property. This is why any approaches to undersand it are very vital.