Results and conjectures on simultaneous core partitions

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

Research output: Contribution to journalArticle

34 Citations (Scopus)

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 - 2014

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Hooks
Partition
Statistics
Catalan number
Abacus
Q-analogue
Combinatorics
Diagram
Major Index
Integer Partitions
Young Diagram
Relatively prime
Affine Group
Coxeter Group
Symmetric group
Statistic
Arrangement
Integer

ASJC Scopus subject areas

  • Geometry and Topology
  • Theoretical Computer Science
  • Computational Theory and Mathematics

Cite this

Results and conjectures on simultaneous core partitions. / Armstrong, Drew; Hanusa, Christopher R H; Jones, Brant C.

In: European Journal of Combinatorics, Vol. 41, 2014, p. 205-220.

Research output: Contribution to journalArticle

Armstrong, Drew ; Hanusa, Christopher R H ; Jones, Brant C. / Results and conjectures on simultaneous core partitions. In: European Journal of Combinatorics. 2014 ; Vol. 41. pp. 205-220.
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