28 research outputs found
A generalisation of a partition theorem of Andrews to overpartitions
In 1969, Andrews proved a theorem on partitions with difference conditions
which generalises Schur's celebrated partition identity. In this paper, we
generalise Andrews' theorem to overpartitions. The proof uses q-differential
equations and recurrences
On Dyson's crank conjecture and the uniform asymptotic behavior of certain inverse theta functions
In this paper we prove a longstanding conjecture by Freeman Dyson concerning
the limiting shape of the crank generating function. We fit this function in a
more general family of inverse theta functions which play a key role in
physics.Comment: Some error bounds have been fixe
An overpartition analogue of the -binomial coefficients
We define an overpartition analogue of Gaussian polynomials (also known as
-binomial coefficients) as a generating function for the number of
overpartitions fitting inside the rectangle. We call these new
polynomials over Gaussian polynomials or over -binomial coefficients. We
investigate basic properties and applications of over -binomial
coefficients. In particular, via the recurrences and combinatorial
interpretations of over q-binomial coefficients, we prove a Rogers-Ramaujan
type partition theorem.Comment: v2: new section added about another way of proving our theorems using
q-series identitie
A generalisation of two partition theorems of Andrews
International audienceIn 1968 and 1969, Andrews proved two partition theorems of the Rogers-Ramanujan type which generalise Schur’s celebrated partition identity (1926). Andrews’ two generalisations of Schur’s theorem went on to become two of the most influential results in the theory of partitions, finding applications in combinatorics, representation theory and quantum algebra. In this paper we generalise both of Andrews’ theorems to overpartitions. The proofs use a new technique which consists in going back and forth from -difference equations on generating functions to recurrence equations on their coefficients.En 1968 et 1969, Andrews a prouvé deux identités de partitions du type Rogers-Ramanujan qui généralisent le célèbre théorème de Schur (1926). Ces deux généralisations sont devenues deux des théorèmes les plus importants de la théorie des partitions, avec des applications en combinatoire, en théorie des représentations et en algèbre quantique. Dans ce papier, nous généralisons les deux théorèmes de Andrews aux surpartitions. Les preuves utilisent une nouvelle technique qui consiste à faire des allers-retours entre équations aux -différences sur les séries génératrices et équations de récurrence sur leurs coefficients
Asymptotic formulae for partition ranks
Using an extension of Wright's version of the circle method, we obtain
asymptotic formulae for partition ranks similar to formulae for partition
cranks which where conjectured by F. Dyson and recently proved by the first
author and K. Bringmann
Characters of level standard modules of as generating functions for generalised partitions
We give a new simple formula for the energy function of a level perfect
crystal of type introduced by Kang, Kashiwara and Misra. We use
this to give several expressions for the characters of level standard
modules as generating functions for different types of partitions. We then
relate one of these formulas to the difference conditions in the conjectural
partition identity of Capparelli, Meurman, Primc and Primc, and prove that
their conjecture is true for all level standard modules. Finally, we
propose a non-specialised generalisation of their conjecture.Comment: 33 page
Bilateral Bailey lattices and Andrews-Gordon type identities
We show that the Bailey lattice can be extended to a bilateral version in
just a few lines from the bilateral Bailey lemma, using a very simple lemma
transforming bilateral Bailey pairs related to into bilateral Bailey pairs
related to . Using this lemma and similar ones, we give bilateral versions
and simple proofs of other (new and known) Bailey lattices, among which a
Bailey lattice of Warnaar and the inverses of Bailey lattices of Lovejoy. As
consequences of our bilateral point of view, we derive new -versions of the
Andrews-Gordon identities, Bressoud's identities, a new companion to Bressoud's
identities, and the Bressoud-G\"ollnitz-Gordon identities. Finally, we give a
new elementary proof of another very general identity of Bressoud using one of
our Bailey lattices.Comment: 27 pages v2: new identities adde
Weighted dependency graphs and the Ising model
Weighted dependency graphs have been recently introduced by the second author, as a toolbox to prove central limit theorems. In this paper, we prove that spins in the d-dimensional Ising model display such a weighted dependency structure. We use this to obtain various central limit theorems for the number of occurrences of local and global patterns in a growing box