Symmetries of k-Symmetric Permutations

I am fascinated by 3-symmetric permutations, that is, permutations that contain all possible patterns of of size three with the same frequency. As I mentioned in my recent post 3-Symmetric Permutations, the smallest non-trivial examples are in size 9.

When I presented these examples at a combinatorics pre-seminar, Sasha Postnikov suggested to draw the permutations as a graph or a matrix. Why didn’t I think of that?

Below are the drawings of the only two 3-symmetric permutations of size 9: 349852167 and 761258943.

3-Symmetric Permutations

As I already mentioned in the aforementioned essay the set of 3-symmetric permutations is invariant under the reversal and subtraction of each number from the size of the permutation plus 1. In geometrical terms it means reflection along the vertical midline and central symmetry. But as you can see the pictures are invariant under 90 degree rotation. Why?

What I forgot to mention was that the set of k-symmetric permutations doesn’t change after the inversion. In geometrical terms it means the reflection with respect to the main diagonal. If you combine a reflection with respect to a diagonal with a reflection with respect to a vertical line you get a 90 degree rotation. Overall, the symmetries of the k-symmetric permutations are the same as all the symmetries of a square. Which means we can only look at the shapes of the k-symmetric permutations.

There are six 2-symmetric permutations: 1432, 2341, 2413, 3142, 3214, 4123. As we can see in the picture below they have two different shapes.

2-Symmetric Permutations

Here is the list of all 22 2-symmetric permutations of size 5: 14532, 15342, 15423, 23541, 24351, 24513, 25143, 25314, 31542, 32451, 32514, 34152, 34215, 35124, 41352, 41523, 42153, 42315, 43125, 51243, 51324, 52134. The list was posted by Drake Thomas in the comments to my essay. Up to symmetries the permutations form four groups. Group 1: 14532, 15423, 23541, 32451, 34215, 43125, 51243, 52134. Group 2: 15342, 24351, 42315, 51324. Group 3: 24513, 25143, 31542, 32514, 34152, 35124, 41523, 42153. Group 4: 25314, 41352. The picture shows the first permutation in each group.

2-Symmetric Permutations of Size 5

One Comment

  1. Angelo Scordo:

    First of all my compliments for your very interisting blog. You have succeeded in transmitting to me also the passion for 3-symmetricic permutations.
    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.

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