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 language | English (US) |
|---|---|
| Pages (from-to) | 205-220 |
| Number of pages | 16 |
| Journal | European Journal of Combinatorics |
| Volume | 41 |
| DOIs | |
| State | Published - 2014 |
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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 journal › Article
}
TY - JOUR
T1 - Results and conjectures on simultaneous core partitions
AU - Armstrong, Drew
AU - Hanusa, Christopher R H
AU - Jones, Brant C.
PY - 2014
Y1 - 2014
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 -