I am wondering whether there exist a name in the literature for the following special type of permutation $\pi$ over $\{1, \dots n\}$. $\pi$ is an involution, i.e., it is composed of nonoverlapping transpositions $(a_i, b_i)$. Now, also the distance between the end-points in each transposition should be the same, i.e., $|a_i - b_i| = k$ for all $i$ for some $k$.
Involutions in which all transpositions have the same distance
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permutations
symmetric-groups
involutions
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1It depends on your purposes. If you just need a name for a secondary concept, it's probably not worth expending much effort to find a suitably evocative term, just something distinctive, such as "(multi-)toggle". If these permutations are your central objects of study, however, readers will welcome a short, memorable term that evokes a visual image closely related to your mathematical aims, e.g.,"track swap" (the ends of railroad ties representing the $a_{i}$ and $b_{i}$, and railroad tracks suggesting constant distance...something like that). – 2017-01-19
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1@AndrewD.Hwang: Thanks for your comment. My question was a bit unclear. I was wondering whether there exist a name in the literature. But if not, these are good suggestions for names to use. – 2017-01-19