One Part Leaky Covers

The paper [7] defined a new class of enumerative invariants called k-leaky double Hurwitz descendants, generalizing both descendant integrals of double ramification cycles and k-leaky double Hurwitz numbers. Here, we focus on the one-part version of these numbers, i.e. when $k\ge 0$ and the positive ramification profile is (d). We derive recursions and use them to produce explicit formulas and structure results for some infinite families of these numbers.