Consider $n$ objects arranged in order: $A=\{1,2,\ldots,n\}$ and an $n$-cycle $\varphi$ from $S_n$.
To motivate what I will ask, consider when $\varphi=(12\ldots n)$, and how the powers of $\varphi$ act on $A$ . For all powers $m$ (mod $n$), $\{1,2,\ldots,n\}^{\varphi^m}$ always has "2" one space to the right of "1", except for once when "2" is $n-1$ spaces to the left of "1". I consider this "bad". I would like the powers of $\varphi$ applied to $A$ to more uniformly distribute the relative positioning of "1" and "2". And not just for "1" and "2" but with any pair "$i$" and "$j$".
Is there a known construction for a $\varphi$ that in some sense optimally distributes these relative positionings?
With $n=3$, there are just two possible $\varphi$, but they are squares of each other so they both fare equally well with respect to what I want to optimize.
With $n=4$, we have $\varphi_1=(1234)$, $\varphi_2=(1243)$, and $\varphi_3=(1324)$ (and their cubes). Applying each one's powers to $\{1,2,3,4\}$ we get:
- $\{1,2,3,4\}$, $\{4,1,2,3\}$, $\{3,4,1,2\}$, $\{2,3,4,1\}$
- $\{1,2,3,4\}$, $\{3,1,4,2\}$, $\{4,3,2,1\}$, $\{2,4,1,3\}$
- $\{1,2,3,4\}$, $\{4,3,1,2\}$, $\{2,1,4,3\}$, $\{3,4,2,1\}$
Item 1. is very bad. "2" is always one space right of "1" (except in one instance). And for example "2" and "4" are never adjacent.
Item 3. has "1" and "2" always adjacent (but at least the switch order half the time).
Item 2. is best:
- Relative to "1", "2" is one to the right, two to the right, one to the left, two to the left. (no repeats)
- Relative to "1", "3" is two to the right, one to the left, two to the left, one to the right. (no repeats)
- Relative to "1", "4" is three to the right, one to the right, three to the left, one to the left. (no repeats)
- Relative to "2", "3" is one to the right, three to the left, one to the left, three to the right. (no repeats)
- Relative to "2", "4" is two to the right, one to the left, two to the left, one to the right. (no repeats)
- Relative to "3", "4" is one to the right, two to the right, one to the left, two to the left. (no repeats)
So by inspection, the answer to my question for $n=4$ is to use $\varphi=(1243)$ (or its cube). But I'd like to know if there is a way to identify a good $\varphi$ for $n$ in general.
