Results and conjectures on simultaneous core partitions

Drew Armstrong, Christopher R.H. Hanusa, Brant C. Jones

Research output: Contribution to journalArticle

47 Scopus citations

Abstract

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. These are related to abacus diagrams and the combinatorics of the affine symmetric group (type A). We observe that self-conjugate 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 (2m n + 1) -core partitions correspond naturally to dominant alcoves in the m-Shi arrangement of type C n, generalizing a result of Fishel-Vazirani for type A. We also introduce a major index statistic on simultaneous n- and (n + 1) -core partitions and on self-conjugate simultaneous 2n- and (2n + 1) -core partitions that yield q-analogs of the Coxeter-Catalan numbers of type A and type C.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.

Original languageEnglish (US)
Pages (from-to)205-220
Number of pages16
JournalEuropean Journal of Combinatorics
Volume41
DOIs
StatePublished - Oct 2014

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

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