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

Core partitions and block coverings

A number of new results about core partitions have been proved recently. ([2],[3], [9], [12]) For s ∈ N an s-core is by definition an integer partition without hooks of length s. This type of partitions first occurred in modular representation theory of symmetric groups, where s-cores label s-blocks of defect 0 in the case where s is a prime. In the study of relations between blocks for differe...

Abaci Structures of (s, ms\pm1)-Core Partitions

We develop a geometric approach to the study of (s,ms−1)-core and (s,ms+1)core partitions through the associated ms-abaci. This perspective yields new proofs for results of H. Xiong and A. Straub (originally proposed by T. Amdeberhan) on the enumeration of (s, s + 1) and (s,ms− 1)-core partitions with distinct parts. It also enumerates the (s,ms+1)-cores with distinct parts. Furthermore, we cal...

T-partitions and S-complete T-partitions of a Graph

Results and conjectures on simultaneous core partitions

An n-core partition is an integer partition whose Young diagram contains no hook lengths equal to n. We consider partitions that are simultaneously a-core and b-core for two relatively prime integers a and b, which correspond to abacus diagrams and the combinatorics of the affine symmetric group (type A). We observe that self-conjugate simultaneous core partitions correspond to type C combinato...

Lattice Points and Simultaneous Core Partitions

We observe that for a and b relatively prime, the “abacus construction” identifies the set of simultaneous (a, b)-core partitions with lattice points in a rational simplex. Furthermore, many statistics on (a, b)-cores are piecewise polynomial functions on this simplex. We apply these results to rational Catalan combinatorics. Using Ehrhart theory, we reprove Anderson’s theorem [3] that there ar...