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

Abstract

AbstractSuppose s and t are coprime positive integers, and let $$\sigma $$ σ be an s-core partition and $$\tau $$ τ a t-core partition. In this paper, we consider the set $${\mathcal {P}}_{\sigma ,\tau }(n)$$ P σ , τ ( n ) of partitions of n with s-core $$\sigma $$ σ and t-core $$\tau $$ τ . We find the smallest n for which this set is non-empty, and show that for this value of n the partitions in $${\mathcal {P}}_{\sigma ,\tau }(n)$$ P σ , τ ( n ) (which we call $$(\sigma ,\tau )$$ ( σ , τ ) -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 $${\mathcal {P}}_{\sigma ,\tau }(n)$$ P σ , τ ( n ) consists of a conjugate pair of partitions, and when $${\mathcal {P}}_{\sigma ,\tau }(n)$$ P σ , τ ( n ) contains a unique self-conjugate partition.