Minimal Partitions with a Given s-Core and t-Core

Abstract

Suppose s and t are coprime positive integers, and let sigma be an s-core partition and tau a t-core partition. In this paper, we consider the set {mathcal {P}}_{sigma ,tau }(n) of partitions of n with s-core sigma and t-core tau . We find the smallest n for which this set is non-empty, and show that for this value of n the partitions in {mathcal {P}}_{sigma ,tau }(n) (which we call (sigma ,tau )-minimal partitions) are in bijection with a certain class of (0, 1)-matrices with s rows and t columns. We then use these results in considering conjugate partitions: we determine exactly when the set {mathcal {P}}_{sigma ,tau }(n) consists of a conjugate pair of partitions, and when {mathcal {P}}_{sigma ,tau }(n) contains a unique self-conjugate partition.