Maximum length and size of simultaneous core partitions with distinct parts

Source abstract

We give, for the first time, a formula for the largest size of an $(s,t)$-core partition with distinct parts valid for every coprime pair. We also determine the greatest number of parts of an $(s,t)$-core partition whose successive parts differ by at least $D$, whenever $\gcd(s,t)\le D$, removing the restrictions on the core parameters in the formula of Burson, Sisneros-Thiry, and Straub. The size formula uses a column description and a single Dedekind sum. The length formula uses coefficients from the Euclidean algorithm and satisfies a recurrence in smaller core parameters. We construct partitions attaining each maximum.