as a vector of complex numbers using as the real part the place i of the digit inside the permutation and as immaginary part the digit itself P(i).

In this way it is easy to calculate the different moment and even the ellipsoid. ]]>

Saying better the concept introduced by my comment of 14 December 2018, the different class in which the permutation group can be divided by

the densities order-isomorphic to the permutations of order 2, can be isomorphic to some properties of the principal moments of inertia applyed

to the permutations.

But the principal moments of inertia and the relevant Poinsot’s Ellipsoid can define better the class subdivision and account for the differences

existing inside the 2-symmetric allowing to identify a subgroup of 2 elemets the “2-supersymmetric permutations” for them the Poinsot’s ellipsoid

is a sphere from the other 4, further investigation show that also for them the ellipsoid is a sphere but its center is not coincident with the center

of gravity. This need furter investigation.

I found the followings:

If you think the points in your graph as having a unitary mass, the center of gravity have coordinates (n/2, n/2).

More interesting is the behaviour of the moment of inetia J.

If we calculate the J with respect to a axis perpendicular to the plane of mass and passing for the center of gravity, the moment of inertia

vary in the same manner as the variance from the 2-symmetric status that I mentioned in my comments to 3-symmetric permutation.

In a first moment I thought that the two measures where isomorphic. But further investigation indicated that the moment of inertia is able to

identify the differences between the 6 2-symmetric permutations, in fact the two central in your graph 15423, 23541 are a perfect distribution

and give the same value of J for every axis passing for the center of gravity. The other four, even if they give the same value with reference

to the vertical and oriziontal axis have a different value for different axis. In fact the moment of inertia in the plane can be represented

as a vector which describe an ellipse with the minimum eccentricity, while the two central make a circle.

On the other end, the permutations: 1234 and 4321 give J=0 if we take an axis passing for the points that represent them. In this case which is

yhe most distant fron the 2-symmetric status, we have that the minimum moment lye on the plane of mass and the maximum is obtained either on

a axis perpendicular to the one passing though the mass or the axis perpendicular to the mass plane and passing for the center of gravity:

This behaviour needs further investigation by a mechanical engineer. ]]>

I want to add something to the 3-symmetric permutations of size 9. I have calculated the distance of all permutation of size 9 from the 3-symmetric status.

I used the variance to measure it. Of course the two 3-simmetric permutation have variance equal to zero. I searched which are the most distant from them, and

found out that they are [1, 2, 3, 4, 5, 6, 7, 8, 9] and [9, 8, 7, 6, 5, 4, 3, 2, 1] with a variance of 980. All the other permutations have intermediate variance from 0 to 980 and the most numerous group has 2328 elements with variance = 35.67. Both, the distribution of variances and the elements of each group are very

irregular. (the previous comment has a calculation error).

Excuse for my poor english, but my mother language is italian.

Thank you in advance.

]]>