41 research outputs found
Results and conjectures on simultaneous core partitions
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 (2mn+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
statistic on simultaneous n- and (n+1)-core partitions and on self-conjugate
simultaneous (2n)- and (2n+1)-core partitions that yield q-analogues 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.Comment: 17 pages; to appear in the European Journal of Combinatoric
Core partitions with distinct parts
Simultaneous core partitions have attracted much attention since Anderson's
work on the number of -core partitions. In this paper we focus on
simultaneous core partitions with distinct parts. The generating function of
-core partitions with distinct parts is obtained. We also prove the results
on the number, the largest size and the average size of -core
partitions. This gives a complete answer to a conjecture of Amdeberhan, which
is partly and independently proved by Straub, Nath and Sellers, and Zaleski
recently.Comment: 8 page
When Does the Set of -Core Partitions Have a Unique Maximal Element?
In 2007, Olsson and Stanton gave an explicit form for the largest -core partition, for any relatively prime positive integers and , and
asked whether there exists an -core that contains all other -cores as subpartitions; this question was answered in the affirmative first
by Vandehey and later by Fayers independently. In this paper we investigate a
generalization of this question, which was originally posed by Fayers: for what
triples of positive integers does there exist an -core
that contains all other -cores as subpartitions? We completely
answer this question when , , and are pairwise relatively prime; we
then use this to generalize the result of Olsson and Stanton.Comment: 8 pages, 2 figure
On the largest sizes of certain simultaneous core partitions with distinct parts
Motivated by Amdeberhan's conjecture on -core partitions with
distinct parts, various results on the numbers, the largest sizes and the
average sizes of simultaneous core partitions with distinct parts were obtained
by many mathematicians recently. In this paper, we derive the largest sizes of
-core partitions with distinct parts, which verifies a
generalization of Amdeberhan's conjecture. We also prove that the numbers of
such partitions with the largest sizes are at most .Comment: 9 page
Theorems, Problems and Conjectures
These notes are designed to offer some (perhaps new) codicils to related
work, a list of problems and conjectures seeking (preferably) combinatorial
proofs. The main items are Eulerian polynomials and hook/contents of Young
diagram, mostly on the latter. The new additions include items on Frobenius
theorem and multi-core partitions; most recently, some problems on (what we
call) colored overpartitions. Formulas analogues to or in the spirit of works
by Han, Nekrasov-Okounkov and Stanley are distributed throughout. Concluding
remarks are provided at the end in hopes of directing the interested
researcher, properly.Comment: 14 page