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 combinatorics, and use abacus diagrams to unite the discussion of these two sets of objects. In particular, we prove that 2nand (2mn+1)-core partitions correspond naturally to dominant alcoves in the m-Shi arrangement of type Cn, generalizing a result of Fishel–Vazirani for type A. We also introduce a major statistic on simultaneous nand (n + 1)-core partitions and on self-conjugate simultaneous 2nand (2n+ 1)-core partitions that yield q-analogues of the type A and type C Coxeter-Catalan numbers. We present related conjectures and open questions on the average size of a simultaneous core partition, q-analogs of generalized Catalan numbers, and generalizations to other Coxeter groups. We also discuss connections with the cyclic sieving phenomenon and q, t-Catalan numbers. To the reader: Section 1 consists of a narrative introducing core partitions, a placement of our results in historical context, and intriguing related conjectures. Section 2 introduces precise definitions of abacus diagrams, which serve as the basis for the proofs of our results. The focus of Section 3 is alcoves in m-Shi arrangements of types A and C. The key result is Theorem 3.5, which characterizes m-minimal and m-bounded regions as a simultaneous core condition, generalizing the result of Fishel and Vazirani [FV10] through a unified method. Theorem 4.4 gives a major statistic on simultaneous core partitions to find a q-analog of the typeA and type C Catalan numbers using abacus diagrams and their bijection with lattice paths; this is the main goal of Section 4. We conclude with a few more open problems motivated by this paper. We hope you enjoy it! 1. CORES AND CONJECTURES A partition of the integer n ∈ N is an unordered multiset of positive integers λ1 ≥ λ2 ≥ · · · ≥ λk > 0 such that ∑k i=1 λi = n. We will write this as λ = (λ1, λ2, . . . , λk) ` n, and say that the size of the partition is n and the length of the partition is k. We will often associate a partition λ with its Young diagram, which is a an array of boxes aligned up and to the left, placing λi boxes in the ith row from the top. For example, here is the Young diagram for the partition (5, 4, 2, 1, 1) ` 13. Date: August 5, 2013. 2010 Mathematics Subject Classification. Primary 05A17, 05A30, 05E15, 20F55; Secondary 05A10, 05A15.