918 research outputs found
Asymptotic formulas for stacks and unimodal sequences
We study enumeration functions for unimodal sequences of positive integers,
where the size of a sequence is the sum of its terms. We survey known results
for a number of natural variants of unimodal sequences, including Auluck's
generalized Ferrer diagrams, Wright's stacks, and Andrews' convex compositions.
These results describe combinatorial properties, generating functions, and
asymptotic formulas for the enumeration functions. We also prove several new
asymptotic results that fill in the notable missing cases from the literature,
including an open problem in statistical mechanics due to Temperley.
Furthermore, we explain the combinatorial and asymptotic relationship between
partitions, Andrews' Frobenius symbols, and stacks with summits.Comment: 19 pages, 4 figure
The Ariki--Koike algebras and Rogers--Ramanujan type partitions
In 2000, Ariki and Mathas showed that the simple modules of the Ariki--Koike
algebras (when
the parameters are roots of unity and ) are labeled by the so-called
Kleshchev multipartitions. This together with Ariki's categorification theorem
enabled Ariki and Mathas to obtain the generating function for the number of
Kleshchev multipartitions by making use of the Weyl--Kac character formula. In
this paper, we revisit this generating function for the case. This
case is particularly interesting, for the corresponding Kleshchev
multipartitions have a very close connection to generalized Rogers--Ramanujan
type partitions when and . Based on
this connection, we provide an analytic proof of the result of Ariki and Mathas
for and . Our second objective is
to investigate simple modules of the Ariki--Koike algebra in a fixed block. It
is known that these simple modules in a fixed block are labeled by the
Kleshchev multiparitions with a fixed partition residue statistic. This
partition statistic is also studied in the works of Berkovich, Garvan, and
Uncu. Employing their results, we provide two bivariate generating function
identities when
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