scholarly journals Continued Classification of 3D Lattice Models in the Positive Octant

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Axel Bacher ◽  
Manuel Kauers ◽  
Rika Yatchak
Keyword(s):  
Asymptotic Behaviour ◽  
Generating Function ◽  
Asymptotic Form ◽  
Three Dimensional ◽  
Lattice Models ◽  
Dimensional Lattice ◽  
Lattice Walks ◽  
Small Set ◽  
International Audience

International audience We continue the investigations of lattice walks in the three-dimensional lattice restricted to the positive octant. We separate models which clearly have a D-finite generating function from models for which there is no reason to expect that their generating function is D-finite, and we isolate a small set of models whose nature remains unclear and requires further investigation. For these, we give some experimental results about their asymptotic behaviour, based on the inspection of a large number of initial terms. At least for some of them, the guessed asymptotic form seems to tip the balance towards non-D-finiteness.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

Low-dimensional lattices. VI. Voronoi reduction of three-dimensional lattices

1992 ◽  
Vol 436 (1896) ◽  
pp. 55-68 ◽  
Keyword(s):  
Simple Formula ◽  
Three Dimensional ◽  
Voronoi Cell ◽  
Dimensional Lattice ◽  
Usual Treatment ◽  
Low Dimensional

The aim of this paper is to describe how the Voronoi cell of a lattice changes as that lattice is continuously varied. The usual treatment is simplified by the introduction of new parameters called the vonorms and conorms of the lattice. The present paper deals with dimensions n ≼ 3; a sequel will treat four-dimensional lattices. An elegant algorithm is given for the Voronoi reduction of a three-dimensional lattice, leading to a new proof of Voronoi’s theorem that every lattice of dimension n ≼ 3 is of the first kind, and of Fedorov’s classification of the three-dimensional lattices into five types. There is a very simple formula for the determinant of a three-dimensional lattice in terms of its conorms.

Solvable Three-Dimensional Lattice Models

1963 ◽  
Vol 132 (3) ◽  
pp. 1085-1092 ◽  
Author(s):  
Bruce W. Knight ◽  
Gerald A. Peterson
Keyword(s):  
Three Dimensional ◽  
Lattice Models ◽  
Dimensional Lattice

Three-Dimensional Lattice Walks in the Upper Half-Space: 10795

2001 ◽  
Vol 108 (10) ◽  
pp. 980 ◽  
Author(s):  
Emeric Deutsch ◽  
Jim Brawner
Keyword(s):  
Half Space ◽  
Three Dimensional ◽  
Dimensional Lattice ◽  
Lattice Walks

scholarly journals Counting quadrant walks via Tutte's invariant method (extended abstract)

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Olivier Bernardi ◽  
Mireille Bousquet-Mélou ◽  
Kilian Raschel
Keyword(s):  
Differential Equation ◽  
Functional Equation ◽  
Linear Differential Equation ◽  
Generating Function ◽  
Algebraic Approach ◽  
Rational Functions ◽  
The Past ◽  
Lattice Walks ◽  
International Audience ◽  
Linear Differential

Extended abstract presented at the conference FPSAC 2016, Vancouver. International audience In the 1970s, Tutte developed a clever algebraic approach, based on certain " invariants " , to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past decade to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic (with one small exception). This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity comes out (almost) automatically. Then, we move to an analytic viewpoint which has already proved very powerful, leading in particular to integral expressions of the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions of the generating function, and a proof that this series is differentially algebraic (that is, satisfies a non-linear differential equation).

Sign in / Sign up

Export Citation Format

Share Document