A new family of weighted double Hurwitz numbers and a new ELSV-type formula with $\mathrm{\Omega }$-classes

We analyze a new family of weighted double Hurwitz numbers that was introduced as a notable example in the context of the $x-y$ duality for logarithmic topological recursion. We use this family to systematically demonstrate, refine and develop techniques that play a crucial role in the interaction of hypergeometric (Orlov–Scherbin) KP tau functions and intersection theory of moduli spaces of curves. In particular, we discuss the subtleties related to the derivation of the ELSV-type formulas in this context and derive a new, explicit ELSV-type formula in terms of the so-called $\mathrm{\Omega }$-classes.