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We give explicit formulas for the number of meromorphic differentials on$\mathbb{CP}^1$ with two zeros and any number of residueless poles and for thenumber of meromorphic differentials on $\mathbb{CP}^1$ with one zero, two poleswith unconstrained residue and any number of residueless poles, in terms of theorders of their zeros and poles. These are the only two finite families ofdifferentials on $\mathbb{CP}^1$ with vanishing residue conditions at a subsetof poles, up to the action of $\mathrm{PGL}(2,\mathbb{C})$. The first family ofnumbers is related to triple Hurwitz numbers by simple integration and we showits connection with the representation theory of $\mathrm{SL}_2(\mathbb{C})$and the equations of the dispersionless KP hierarchy. The second family has avery simple generating series, and we recover it through surprisingly involvedcomputations using intersection theory of moduli spaces of curves anddifferentials.

Counting meromorphic differentials on $\mathbb{CP}^1$