A Chinese Remainder Theorem for partitions

Abstract

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