501 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
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
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
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
Polynomial equations with one catalytic variable, algebraic series, and map enumeration
Let be a formal power series in with polynomial
coefficients in . Let be formal power series in ,
independent of . Assume all these series are characterized by a polynomial
equation We prove that, under a mild
hypothesis on the form of this equation, these series are algebraic,
and we give a strategy to compute a polynomial equation for each of them. This
strategy generalizes the so-called kernel method, and quadratic method, which
apply respectively to equations that are linear and quadratic in .
Applications include the solution of numerous map enumeration problems, among
which the hard-particle model on general planar maps
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
- …