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


[Cyclic orderings](https://en.wikipedia.org/wiki/Cyclic_order) 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.



Weights on cyclic orderings
cyclic permutations, and a need to define distances on them, pops up in literature on graph crossing numbers, e.g. we used it in <http://arxiv.org/abs/math/0404142> 


It probably goes back all the way to at least 


D.J. Kleitman, The crossing number of $K\_5$. J. Combinatorial Theory 9 (1970), 315–323.


