Completed Cycles Leaky Hurwitz Numbers

Source abstract

Abstract We introduce $(r+1)$-completed cycles $k$-leaky Hurwitz numbers and prove piecewise polynomiality as well as establishing their chamber polynomiality structure and their wall crossing formulae. For $k=0$ the results recover previous results of Shadrin–Spitz–Zvonkine. The specialization for $r=1$ recovers Hurwitz numbers that are close to the ones studied by Cavalieri–Markwig–Ranganathan and Cavalieri–Markwig–Schmitt.