Lattice points and simultaneous core partitions

We apply lattice point techniques to the study of simultaneous core partitions.Our central observation is that for a and b relatively prime, the abacus construction identifies the set of simultaneous (a, b)-core partitions with lattice points in a rational simplex. We apply this result in two main ways: using Ehrhart theory, we reprove Anderson’s theorem that there are (a+b−1)!/a!b! simultaneous (a, b)-cores; and using Euler-Maclaurin theory we prove Armstrong’s conjecture that the average sizeof an (a, b)-core is (a+b+1)(a−1)(b−1)/24. Our methods also give new derivations of analogous formulas for the number and average size of self-conjugate (a, b)-cores.