Conjugation in Sn

This note is in two parts. The first part contains proofs of four elementary criteria for two permutations in the symmetric group [Formula: see text] to be conjugate in [Formula: see text]. Most readers know the first criterion but perhaps fewer are familiar with the remaining three. The second part contains a discussion of the problem of deciding whether two ordered pairs of permutations are conjugate by an element in [Formula: see text]. In sharp contrast to the situation in part one, here no elementary criteria are known. However, pairs of permutations give rise to dessins d’enfants and two ordered pairs of permutations are conjugate precisely when the corresponding dessins are isomorphic. We give an expository discussion, with examples and references but no proofs, of some of this material; we hope our discussion will help readers who are encountering this known material for the first time.