A Chinese Remainder Theorem for partitions

Abstract

Let s, t be natural numbers and fix an s-core partition sigma and a t-core partition tau. Put d = gcd(s, t) and m = lcm(s, t), and write N-sigma,N-tau(k) for the number of m-core partitions of length no greater than k whose s-core is sigma and t-core is tau. We prove that for k large, N-sigma,N-tau (k) is a quasipolynomial of period m and degree 1/d (s - d)(t - d).