The “classical” parking functions, counted by the Cayley number (n+ 1) n− 1, carry a natural
permutation representation of the symmetric group S n in which the number of orbits is the …

We give a new characterization of the vertical-strip LLT polynomials GP (x; q) as the unique
family of symmetric functions that satisfy certain combinatorial relations. This …

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 …

A special case of an elegant result due to Anderson proves that the number of (s, s+ 1)-core
partitions is finite and is given by the Catalan number C s. Amdeberhan recently conjectured …

Hooks are prominent in representation theory (of symmetric groups) and they play a role in
number theory (via cranks associated to Ramanujan's congruences). A partition of a positive …

t-core partitions have played important roles in the theory of partitions and related areas. In
this survey, we briefly summarize interesting and important results on t-cores from classical …

Fix coprime $ s, t\ge1 $. We re-prove, without Ehrhart reciprocity, a conjecture of Armstrong
(recently verified by Johnson) that the finitely many simultaneous $(s, t) $-cores have …

A beautiful recent conjecture of Armstrong predicts the average size of a partition that is
simultaneously an s-core and a t-core, where s and t are coprime. Our goal is to prove this …

We define a family of maps on lattice paths, called sweep maps, that assign levels to each
step in the path and sort steps according to their level. Surprisingly, although sweep maps …

Anderson established a connection between core partitions and order ideals of certain
posets by mapping a partition to its β-set. In this paper, we give a description of the posets P …