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:


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.

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    $\begingroup$ See also mathoverflow.net/questions/58782/… - Woodall's "Cyclic-order graphs and Zarankiewicz's crossing-number conjecture" is the most interesting paper on this, from my perspective, because it actually constructs the graph of these things under a certain definition of this distance (one I like best). I do not know of any theoretical discussion of the issue you mention, though I'm glad to hear of Dima's computational one - and if it was actually more theoretical in that paper than I though, he'll let me know :) which I will find useful. $\endgroup$
    – kcrisman
    Oct 27, 2015 at 19:03
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    $\begingroup$ we (and Woodall, and Kleitman) define the graph on these $(n-1)!$ elements of $S_n$, so that two such elements are adjacent if one needs to swap two adjacent positions to get one from the other. $\endgroup$ Oct 27, 2015 at 22:10

1 Answer 1


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.


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