Are there standard or known weights/metrics on cyclic orders?

Cyclic orderings are different ways of listing elements from a finite set, where you call two lists the same if they differ only by a rotation. For example, the ways that 4 people can sit at a circular table, where you ignore rotation of the table.

So the following 4 rankings of a 4-item set should be considered the same:

```
[2,1,4,3]
[1,4,3,2]
[4,3,2,1]
[3,2,1,4]
```

You can think of this as arranging the 4 items on a circular list, with no particular top, but a definite direction.

I would like to compare different cyclic orderings in a meaningful way. I have learned that there are many well-known ways of assigning distance or weight to non-cyclic orderings (a.k.a. permutations!). For example:

- Kendall $\tau$
- Transposition distance (Cayley)
- Ulam metric
- Hamming weight
- $\ell_1$ norm

But these give different weights to permutations which correspond to the same cyclic ordering.

A first simple idea is the following: given a permutation $a$, count the number of $i$ such that $a(i+1) \neq a(i) +1$, where $0 \leq i \leq n$ and index addition is performed modulo $n$.

A natural variation on this idea would sum $|a(i+1) - (a(i)+1)|$.

I haven't thought about either of these too carefully yet. Do they appear in the literature somewhere? Some nontrivial internet searching hasn't turned anything up.

theoreticaldiscussion of the issue you mention, though I'm glad to hear of Dima'scomputationalone - and if it was actually more theoretical in that paper than I though, he'll let me know :) which I will find useful. $\endgroup$