Circular sorting in the alternating group

Abstract

The symmetric group $S_n$ is generated by transpositions, and problems of sorting permutations using transpositions are well studied. In recent work, Adin, Alon, and Roichman studied the related problem of sorting $n$ points on a circle, and gave a formula for the maximum number of adjacent swaps required. This is equivalent to the number of adjacent transpositions required to transform any permutation into a power of the cyclic permutation $(1,2,\ldots, n)$. The focus of this work is an analogous question in the alternating group $A_n$, which is generated by $3$-cycles. That is, using 3-cycles instead of transpositions, what is the maximum number of steps required to transform an even permutation into a power of $(1,2,\ldots, n)$ in the alternating group? We determine this number exactly for even $n$ and $n \equiv 1 \pmod{4}$. For $n \equiv 3 \pmod{4}$, we show that the sorting number can take one of two possible values and give explicit constructions demonstrating that the larger value occurs infinitely often.