The shallow permutations are the unlinked permutations

Abstract: Diaconis and Graham studied a measure of distance from the identity in the symmetric group called total displacement and showed that it is bounded below by the sum of length and reflection length. They asked for a characterization of the permutations where this bound is an equality; we call these the shallow permutations. Cornwell and McNew recently interpreted the cycle diagram of a permutation as a knot diagram and studied the set of permutations for which the corresponding link is an unlink. We show the shallow permutations are precisely the unlinked permutations. As Cornwell and McNew give a generating function counting unlinked permutations, this gives a generating function counting shallow permutations.