TY - JOUR
T1 - Results and conjectures on simultaneous core partitions
AU - Armstrong, Drew
AU - Hanusa, Christopher R.H.
AU - Jones, Brant C.
N1 - Funding Information:
D. Armstrong was partially supported by NSF award DMS-1001825 . C.R.H. Hanusa gratefully acknowledges support from PSC-CUNY Research Awards TRADA-43-127 and TRADA-44-168 . B.C. Jones was supported in part by a CSM Summer Research Grant from James Madison University .
PY - 2014/10
Y1 - 2014/10
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84899903183&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84899903183&partnerID=8YFLogxK
U2 - 10.1016/j.ejc.2014.04.007
DO - 10.1016/j.ejc.2014.04.007
M3 - Article
AN - SCOPUS:84899903183
VL - 41
SP - 205
EP - 220
JO - European Journal of Combinatorics
JF - European Journal of Combinatorics
SN - 0195-6698
ER -