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We prove the quasimodularity of generating functions for counting pillowcasecovers, with and without Siegel-Veech weight. Similar to prior work on toruscovers, the proof is based on analyzing decompositions of half-translationsurfaces into horizontal cylinders. It provides an alternative proof of thequasimodularity results of Eskin-Okounkov and a practical method to computearea Siegel-Veech constants. A main new technical tool is a quasi-polynomialityresult for 2-orbifold Hurwitz numbers with completed cycles.

Pillowcase covers: Counting Feynman-like graphs associated with quadratic differentials