A generalisation of two partition theorems of Andrews

International audience In 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 $q$-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 $q$-différences sur les séries génératrices et équations de récurrence sur leurs coefficients.

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On covariance generating functions and spectral densities of periodically correlated autoregressive processes

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/jamsa/2006/94746 ◽

2006 ◽

Vol 2006 ◽

pp. 1-17 ◽

Cited By ~ 4

Author(s):

Z. Shishebor ◽

A. R. Nematollahi ◽

A. R. Soltani

Keyword(s):

Finite Order ◽

Spectral Theory ◽

Generating Functions ◽

New Technique ◽

Autoregressive Processes ◽

Nonstationary Processes ◽

Spectral Densities ◽

Covariance Functions ◽

Periodically Correlated Processes ◽

A New Technique

Periodically correlated autoregressive nonstationary processes of finite order are considered. The corresponding Yule-Walker equations are applied to derive the generating functions of the covariance functions, what are called here the periodic covariance generating functions. We also provide closed formulas for the spectral densities by using the periodic covariance generating functions, which is a new technique in the spectral theory of periodically correlated processes.

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Enumerating projective reflection groups

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2898 ◽

2011 ◽

Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽

Author(s):

Riccardo Biagioli ◽

Fabrizio Caselli

Keyword(s):

Representation Theory ◽

Generating Functions ◽

Special Class ◽

Hilbert Series ◽

Weyl Groups ◽

Reflection Groups ◽

One Dimensional ◽

Complex Reflection Groups ◽

Major Index ◽

International Audience

International audience Projective reflection groups have been recently defined by the second author. They include a special class of groups denoted G(r,p,s,n) which contains all classical Weyl groups and more generally all the complex reflection groups of type G(r,p,n). In this paper we define some statistics analogous to descent number and major index over the projective reflection groups G(r,p,s,n), and we compute several generating functions concerning these parameters. Some aspects of the representation theory of G(r,p,s,n), as distribution of one-dimensional characters and computation of Hilbert series of some invariant algebras, are also treated. Les groupes de réflexions projectifs ont été récemment définis par le deuxième auteur. Ils comprennent une classe spéciale de groupes notée G(r,p,s,n), qui contient tous les groupes de Weyl classiques et plus généralement tous les groupes de réflexions complexes du type G(r,p,n). Dans ce papier on définit des statistiques analogues au nombre de descentes et à l'indice majeur pour les groupes G(r,p,s,n), et on calcule plusieurs fonctions génératrices. Certains aspects de la théorie des représentations de G(r,p,s,n), comme la distribution des caractères linéaires et le calcul de la série de Hilbert de quelques algèbres d'invariants, sont aussi abordés.

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Constructing neighborly polytopes and oriented matroids

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3032 ◽

2012 ◽

Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽

Author(s):

Arnau Padrol

Keyword(s):

Oriented Matroids ◽

New Technique ◽

Oriented Matroid ◽

Nous Proposons ◽

International Audience ◽

New Construction ◽

A New Technique ◽

Neighborly Polytope

International audience A $d$-polytope $P$ is neighborly if every subset of $\lfloor\frac{d}{2}\rfloor $vertices is a face of $P$. In 1982, Shemer introduced a sewing construction that allows to add a vertex to a neighborly polytope in such a way as to obtain a new neighborly polytope. With this, he constructed superexponentially many different neighborly polytopes. The concept of neighborliness extends naturally to oriented matroids. Duals of neighborly oriented matroids also have a nice characterization: balanced oriented matroids. In this paper, we generalize Shemer's sewing construction to oriented matroids, providing a simpler proof. Moreover we provide a new technique that allows to construct balanced oriented matroids. In the dual setting, it constructs a neighborly oriented matroid whose contraction at a particular vertex is a prescribed neighborly oriented matroid. We compare the families of polytopes that can be constructed with both methods, and show that the new construction allows to construct many new polytopes. Un $d$-polytope $P$ est $\textit{neighborly}$ si tout sous-ensemble de $\lfloor\frac{d}{2}\rfloor $ sommets forme une face de $P$. En 1982, Shemer a introduit une construction de couture qui permet de rajouter un sommet à un polytope $\textit{neighborly}$ et d'obtenir un nouveau polytope $\textit{neighborly}$. Cette construction lui permet de construire un nombre super-exponentiel de polytopes $\textit{neighborly}$ distincts. Le concept de $\textit{neighborliness}$ s'étend naturellement aux matroïdes orientés. Les duaux de matroïdes orientés $\textit{neighborly}$ ont de plus une belle caractérisation: ce sont les matroïdes orientés équilibrés. Dans cet article, nous généralisons la construction de couture de Shemer aux matroïdes orientés, ce qui en fournit une démonstration plus simple. Par ailleurs, nous proposons une nouvelle technique qui permet de construire matroïdes orientés équilibrés. Dans le cadre dual, on obtient un matroïde $\textit{neighborly}$ dont la contraction à un sommet distinguè est un matroïde $\textit{neighborly}$ prescrit. Nous comparons les familles de polytopes qui peuvent être construites avec ces deux méthodes, et montrons que la nouvelle construction permet de construire plusieurs nouveaux polytopes.

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Bounds for d-distinct partitions

10.46298/hrj.2021.7430 ◽

2021 ◽

Vol Volume 43 - Special... ◽

Author(s):

Soon-Yi Kang ◽

Young Kim

Keyword(s):

Upper Bound ◽

Generating Functions ◽

Upper Bounds ◽

Modular Functions ◽

Lower And Upper Bounds ◽

Partition Identity ◽

Recent Developments ◽

International Audience ◽

Partition Inequalities ◽

Euler’S Identity

International audience Euler's identity and the Rogers-Ramanujan identities are perhaps the most famous results in the theory of partitions. According to them, 1-distinct and 2-distinct partitions of n are equinumerous with partitions of n into parts congruent to ±1 modulo 4 and partitions of n into parts congruent to ±1 modulo 5, respectively. Furthermore, their generating functions are modular functions up to multiplication by rational powers of q. For d ≥ 3, however, there is neither the same type of partition identity nor modularity for d-distinct partitions. Instead, there are partition inequalities and mock modularity related with d-distinct partitions. For example, the Alder-Andrews Theorem states that the number of d-distinct partitions of n is greater than or equal to the number of partitions of n into parts which are congruent to ±1 (mod d+3). In this note, we present the recent developments of generalizations and analogs of the Alder-Andrews Theorem and establish asymptotic lower and upper bounds for the d-distinct partitions. Using the asymptotic relations and data obtained from computation, we propose a conjecture on a partition inequality that gives an upper bound for d-distinct partitions. Specifically, for d ≥ 4, the number of d-distinct partitions of n is less than or equal to the number of partitions of n into parts congruent to ±1 (mod m), where m ≤ 2dπ^2 / [3 log^2 (d)+6 log d] .

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