Ehrhart functions and symplectic embeddings of ellipsoids

McDuff has previously shown that one four‐dimensional symplectic ellipsoid can be symplectically embedded into another if and only if a certain combinatorial criteria holds. We reinterpret this combinatorial criteria using the theory of Ehrhart quasipolynomials, and we use this to give purely combinatorial proofs of theorems of McDuff–Schlenk and Frenkel–Müller, concerning the existence of ‘infinite staircases’ in symplectic embedding problems. We then find a third, new, staircase and conjecture that these are the only three staircases for embeddings into rational ellipsoids. Several other applications are also discussed; for example, we give new examples of triangles whose Ehrhart function exhibits a period collapse.