Centrality of star and monotone factorisations

Abstract

A factorisation problem in the symmetric group is central if conjugate permutations always have the same number of factorisations. We give the first fully combinatorial proof of the centrality of transitive star factorisations that is valid in all genera, which answers a natural question of Goulden and Jackson from 2009. We begin by showing that the set of star factorisations is equinumerous with a certain set of monotone factorisations, a new result. We give more than one proof of this, and, crucially, one of our proofs is bijective. As a corollary we obtain new formulae for some monotone double Hurwitz factorisations, and a new relation between Hurwitz and monotone Hurwitz factorisations. We also generalise a theorem of Goulden and Jackson from 2009 that states that the transitive power of Jucys-Murphy elements are central. Our theorem states that the transitive image of any symmetric function evaluated at Jucys-Murphy elements is central, which gives a transitive version of Jucys' original result from 1974.