Computing triple and double Hurwitz numbers involving a branch point with a two-part profile

Abstract

The study of Hurwitz numbers intersects with many research areas including representation theory, algebraic geometry and mathematical physics. Though many beautiful general properties have been discovered, obtaining explicit elementary expressions computing these numbers is hard and pertains to a primary goal of the topic. In fact, known explicit formulas are mainly for Hurwitz numbers involving at most two nonsimple branch points (i.e., double Hurwitz numbers). Even for double Hurwitz numbers, only the case where one of the branch points is fully ramified (i.e., one-part double Hurwitz numbers) has been completely and explicitly determined. In this paper, we contribute explicit elementary formulas computing Hurwitz numbers with completed r-cycles involving up to three nonsimple branch points where one of them has a two-part profile, enriching several lines of researches. In particular, we discuss the piecewise polynomiality and the genus-zero case in detail.