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.