Results and conjectures on simultaneous core partitions
Abstract
An ncore partition is an integer partition whose Young diagram contains no hook lengths equal to n. We consider partitions that are simultaneously acore and bcore for two relatively prime integers a and b. These are related to abacus diagrams and the combinatorics of the affine symmetric group (type A). We observe that selfconjugate simultaneous core partitions correspond to the combinatorics of type C, and use abacus diagrams to unite the discussion of these two sets of objects. In particular, we prove that (2n) and (2mn+1)core partitions correspond naturally to dominant alcoves in the mShi arrangement of type C_n, generalizing a result of FishelVazirani for type A. We also introduce a major statistic on simultaneous n and (n+1)core partitions and on selfconjugate simultaneous (2n) and (2n+1)core partitions that yield qanalogues of the CoxeterCatalan numbers of type A and type C. We present related conjectures and open questions on the average size of a simultaneous core partition, qanalogs of generalized Catalan numbers, and generalizations to other Coxeter groups. We also discuss connections with the cyclic sieving phenomenon and q,tCatalan numbers.
 Publication:

arXiv eprints
 Pub Date:
 August 2013
 arXiv:
 arXiv:1308.0572
 Bibcode:
 2013arXiv1308.0572A
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 17 pages