Pairing conjugate partitions by residue classes

The Number of Parts in Certain Residue Classes of Integer Partitions

We use the Circle Method to derive asymptotics for functions related to the number of parts of partitions in particular residue classes.

Cranks and T -cores

New statistics on partitions (called cranks) are defined which combinatorially prove Ramanujan’s congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given for the equinumerous crank classes. The cranks are closely related to the t-core of a partition. Using q-series, some explicit formulas are given for the number of partitions which are t-cores. Some related...

Dominant residue classes concerning the summands of partitions

On the Number of Parts of Integer Partitions Lying in given Residue Classes

Improving upon previous work [3] on the subject, we use Wright’s Circle Method to derive an asymptotic formula for the number of parts in all partitions of an integer n that are in any given arithmetic progression.

Lecture hall sequences, q-series, and asymmetric partition identities

We use generalized lecture hall partitions to discover a new pair of q-series identities. These identities are unusual in that they involve partitions into parts from asymmetric residue classes, much like the little Göllnitz partition theorems. We derive a two-parameter generalization of our identities that, surprisingly, gives new analytic counterparts of the little Göllnitz theorems. Finally,...