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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.

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  • $\begingroup$ The numbers you described are know oeis.org/A000140. But what is the question? (As of now it sound open-ended.) $\endgroup$ – Piotr Migdal Aug 7 '14 at 14:59
  • $\begingroup$ Piotr Migdal: Thank you. A few years ago the $n+1/2$ property of $T(n,k)$ was found just by pure accident ( I would say). I hoped to find something similar else via a geometry approach, but failed. Moreover, the generation function for inversion and formula for Products of necklaces (one of my other questions at MO) seems to be connected. So, I am trying to understand why it happens? $\endgroup$ – Mikhail Gaichenkov Aug 7 '14 at 16:28
  • $\begingroup$ Piotr Migdal: Why the $n+1/2$ appears in physics and what is behind it if we would apply the property of numbers to quantum physics models (or extend the model). $\endgroup$ – Mikhail Gaichenkov Aug 7 '14 at 16:35

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