scholarly journals Extended Rate, more GFUN

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Martin Rubey
Keyword(s):  
Software Package ◽  
Rational Functions ◽  
Rational Numbers ◽  
International Audience

International audience We present a software package that guesses formulas for sequences of, for example, rational numbers or rational functions, given the first few terms. Thereby we extend and complement Christian Krattenthaler’s program $\mathtt{Rate}$ and the relevant parts of Bruno Salvy and Paul Zimmermann’s $\mathtt{GFUN}$.

scholarly journals Operations on partially ordered sets and rational identities of type A

2013 ◽  
Vol Vol. 15 no. 2 (Combinatorics) ◽  
Author(s):  
Adrien Boussicault
Keyword(s):  
Partially Ordered Sets ◽  
Rational Functions ◽  
Hasse Diagram ◽  
Linear Extensions ◽  
Ordered Sets ◽  
Closed Expression ◽  
Schubert Polynomials ◽  
Special Cases ◽  
The Family ◽  
International Audience

Combinatorics International audience We consider the family of rational functions ψw= ∏( xwi - xwi+1 )-1 indexed by words with no repetition. We study the combinatorics of the sums ΨP of the functions ψw when w describes the linear extensions of a given poset P. In particular, we point out the connexions between some transformations on posets and elementary operations on the fraction ΨP. We prove that the denominator of ΨP has a closed expression in terms of the Hasse diagram of P, and we compute its numerator in some special cases. We show that the computation of ΨP can be reduced to the case of bipartite posets. Finally, we compute the numerators associated to some special bipartite graphs as Schubert polynomials.

Series representing transcendental numbers that are not U-numbers

2015 ◽  
Vol 11 (03) ◽  
pp. 869-892
Author(s):  
Emre Alkan
Keyword(s):  
Rational Functions ◽  
Rational Numbers ◽  
Upper Bounds ◽  
Integral Representations ◽  
Binomial Coefficients ◽  
Elementary Functions ◽  
Linear Combinations ◽  
Type Series ◽  
Transcendental Numbers ◽  
Mathematical Constants

Using integral representations with carefully chosen rational functions as integrands, we find new families of transcendental numbers that are not U-numbers, according to Mahler's classification, represented by a series whose terms involve rising factorials and reciprocals of binomial coefficients analogous to Apéry type series. Explicit descriptions of these numbers are given as linear combinations with coefficients lying in a suitable real algebraic extension of rational numbers using elementary functions evaluated at arguments belonging to the same field. In this way, concrete examples of transcendental numbers which can be expressed as combinations of classical mathematical constants such as π and Baker periods are given together with upper bounds on their wn measures.

Counting points of given height that generate a quadratic extension of a function field

2015 ◽  
Vol 11 (02) ◽  
pp. 569-592 ◽  
Author(s):  
David Kettlestrings ◽  
Jeffrey Lin Thunder
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Function Field ◽  
Asymptotic Estimate ◽  
Algebraic Closure ◽  
Quadratic Extension ◽  
Rational Functions ◽  
Rational Numbers ◽  
Algebraic Extension ◽  
Counting Points

Let K be a finite algebraic extension of the field of rational functions in one indeterminate over a finite field and let [Formula: see text] denote an algebraic closure of K. We count points in projective space [Formula: see text] with given height and generating a quadratic extension of K. If n > 2, we derive an asymptotic estimate for the number of such points as the height tends to infinity. Such estimates are analogous to previous results of Schmidt where the field K is replaced by the field of rational numbers ℚ.

scholarly journals Perfect Matchings for the Three-Term Gale-Robinson Sequences

10.37236/214 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Mireille Bousquet-Mélou ◽  
James Propp ◽  
Julian West
Keyword(s):  
Recurrence Relations ◽  
Initial Conditions ◽  
A Priori ◽  
Rational Functions ◽  
Rational Numbers ◽  
Perfect Matchings ◽  
Parameter Family ◽  
Integer Coefficients ◽  
Special Cases

In 1991, David Gale and Raphael Robinson, building on explorations carried out by Michael Somos in the 1980s, introduced a three-parameter family of rational recurrence relations, each of which (with suitable initial conditions) appeared to give rise to a sequence of integers, even though a priori the recurrence might produce non-integral rational numbers. Throughout the '90s, proofs of integrality were known only for individual special cases. In the early '00s, Sergey Fomin and Andrei Zelevinsky proved Gale and Robinson's integrality conjecture. They actually proved much more, and in particular, that certain bivariate rational functions that generalize Gale-Robinson numbers are actually polynomials with integer coefficients. However, their proof did not offer any enumerative interpretation of the Gale-Robinson numbers/polynomials. Here we provide such an interpretation in the setting of perfect matchings of graphs, which makes integrality/polynomiality obvious. Moreover, this interpretation implies that the coefficients of the Gale-Robinson polynomials are positive, as Fomin and Zelevinsky conjectured.

scholarly journals Cumulants of Jack symmetric functions and b-conjecture (extended abstract)

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Maciej Dolega ◽  
Valentin Féray
Keyword(s):  
Nonnegative Integer ◽  
Symmetric Functions ◽  
Rational Functions ◽  
Independent Interest ◽  
Jack Polynomials ◽  
Factorization Property ◽  
Integer Coefficients ◽  
Continuous Deformation ◽  
Generating Series ◽  
International Audience

International audience Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t, 1 + β) that might be interpreted as a continuous deformation of the rooted hypermap generating series. They made the following conjecture: coefficients of ψ(x, y, z; t, 1+β) are polynomials in β with nonnegative integer coefficients. We prove partially this conjecture, nowadays called b-conjecture, by showing that coefficients of ψ(x, y, z; t, 1 + β) are polynomials in β with rational coefficients. Until now, it was only known that they are rational functions of β. A key step of the proof is a strong factorization property of Jack polynomials when α → 0 that may be of independent interest.

scholarly journals Generalized Ehrhart polynomials

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Sheng Chen ◽  
Nan Li ◽  
Steven V Sam
Keyword(s):  
Diophantine Equations ◽  
Rational Functions ◽  
Lattice Points ◽  
Classical Theorem ◽  
Number Of Solutions ◽  
Ehrhart Polynomials ◽  
Linear Diophantine Equations ◽  
International Audience

International audience Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations $P(n) = nP$ is a quasi-polynomial in $n$. We generalize this theorem by allowing the vertices of $P(n)$ to be arbitrary rational functions in $n$. In this case we prove that the number of lattice points in $P(n)$ is a quasi-polynomial for $n$ sufficiently large. Our work was motivated by a conjecture of Ehrhart on the number of solutions to parametrized linear Diophantine equations whose coefficients are polynomials in $n$, and we explain how these two problems are related. Soit $P$ un polytope avec sommets rationelles. Un théorème classique des Ehrhart déclare que le nombre de points du réseau dans les dilatations $P(n) = nP$ est un quasi-polynôme en $n$. Nous généralisons ce théorème en permettant à des sommets de $P(n)$ comme arbitraire fonctions rationnelles en $n$. Dans ce cas, nous prouvons que le nombre de points du réseau en $P(n)$ est une quasi-polynôme pour $n$ assez grand. Notre travail a été motivée par une conjecture d'Ehrhart sur le nombre de solutions à linéaire paramétrée Diophantine équations dont les coefficients sont des polyômes en $n$, et nous expliquer comment ces deux problèmes sont liés.

scholarly journals Arc-Coloured Permutations

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Lily Yen
Keyword(s):  
Generating Functions ◽  
Joint Distribution ◽  
Rational Functions ◽  
Set Partitions ◽  
Combinatorial Objects ◽  
Crossing Numbers ◽  
Rational Series ◽  
Nous Montrons ◽  
International Audience ◽  
Labelled Graphs

International audience The equidistribution of many crossing and nesting statistics exists in several combinatorial objects like matchings, set partitions, permutations, and embedded labelled graphs. The involutions switching nesting and crossing numbers for set partitions given by Krattenthaler, also by Chen, Deng, Du, Stanley, and Yan, and for permutations given by Burrill, Mishna, and Post involved passing through tableau-like objects. Recently, Chen and Guo for matchings, and Marberg for set partitions extended the result to coloured arc annotated diagrams. We prove that symmetric joint distribution continues to hold for arc-coloured permutations. As in Marberg's recent work, but through a different interpretation, we also conclude that the ordinary generating functions for all j-noncrossing, k-nonnesting, r-coloured permutations according to size n are rational functions. We use the interpretation to automate the generation of these rational series for both noncrossing and nonnesting coloured set partitions and permutations. <begin>otherlanguage*</begin>french L'équidistribution de plusieurs statistiques décrites en termes d'emboitements et de chevauchements d'arcs s'observes dans plusieurs familles d'objects combinatoires, tels que les couplages, partitions d'ensembles, permutations et graphes étiquetés. L'involution échangeant le nombre d'emboitements et de chevauchements dans les partitions d'ensemble due à Krattenthaler, et aussi Chen, Deng, Du, Stanley et Yan, et l'involution similaire dans les permutations due à Burrill, Mishna et Post, requièrent d'utiliser des objets de type tableaux. Récemment, Chen et Guo pour les couplages, et Marberg pour les partitions d'ensembles, ont étendu ces résultats au cas de diagrammes arc-annotés coloriés. Nous démontrons que la propriété d'équidistribution s'observe est aussi vraie dans le cas de permutations aux arcs coloriés. Tout comme dans le travail résent de Marberg, mais via un autre chemin, nous montrons que les séries génératrices ordinaires des permutations r-coloriées ayant au plus j chevauchements et k emboitements, comptées selon la taille n, sont des fonctions rationnelles. Nous décrivons aussi des algorithmes permettant de calculer ces fonctions rationnelles pour les partitions d'ensembles et les permutations coloriées sans emboitement ou sans chevauchement. <end>otherlanguage*</end>

scholarly journals Asymptotics of lattice walks via analytic combinatorics in several variables

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Stephen Melczer ◽  
Mark C. Wilson
Keyword(s):  
Generating Functions ◽  
Kernel Method ◽  
Rational Functions ◽  
Two Dimensional ◽  
Combinatorial Properties ◽  
Analytic Combinatorics ◽  
Several Variables ◽  
Algebra Approach ◽  
Finite Set ◽  
International Audience

International audience We consider the enumeration of walks on the two-dimensional non-negative integer lattice with steps defined by a finite set S ⊆ {±1, 0}2 . Up to isomorphism there are 79 unique two-dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Pade ́-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.

scholarly journals A Closed Character Formula for Symmetric Powers of Irreducible Representations

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Stavros Kousidis
Keyword(s):  
Irreducible Representation ◽  
Rational Functions ◽  
Character Formula ◽  
Irreducible Representations ◽  
Partition Functions ◽  
Partial Fraction ◽  
Residue Type ◽  
Symmetric Powers ◽  
International Audience ◽  
Partial Fraction Decomposition

International audience We prove a closed character formula for the symmetric powers $S^N V(λ )$ of a fixed irreducible representation $V(λ )$ of a complex semi-simple Lie algebra $\mathfrak{g}$ by means of partial fraction decomposition. The formula involves rational functions in rank of $\mathfrak{g}$ many variables which are easier to determine than the weight multiplicities of $S^N V(λ )$ themselves. We compute those rational functions in some interesting cases. Furthermore, we introduce a residue-type generating function for the weight multiplicities of $S^N V(λ )$ and explain the connections between our character formula, vector partition functions and iterated partial fraction decomposition. Nous établissons une formule fermée pour le caractère de la puissance symétrique $S^N V(λ )$ d'une représentation irréductible $V(λ )$ d'une algèbre de Lie semi-simple complexe$\mathfrak{g}$, en utilisant des décompositions en fractions partielles. Cette formule exprime ce caractère en termes de fractions rationnelles en $r$ variables, où $r$ est le rang de $\mathfrak{g}$. Ces fractions sont plus faciles à déterminer que les multiplicités de la décomposition de $S^N V(λ )$ elles-mêmes. Nous calculons ces fonctions rationnelles dans quelques cas intéressants. Nous introduisons par ailleurs une fonction génératrice de type résidu pour les multiplicités de $S^N V(λ )$ et relions notre formule aux fonctions de partitions vectorielles et aux décompositions itérées en fractions partielles.

scholarly journals Structure of spanning trees on the two-dimensional Sierpinski gasket

2011 ◽  
Vol Vol. 12 no. 3 (Combinatorics) ◽  
Author(s):  
Shu-Chiuan Chang ◽  
Lung-Chi Chen
Keyword(s):  
Probability Distribution ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Sierpinski Gasket ◽  
Rational Numbers ◽  
Limiting Distribution ◽  
Two Dimensional ◽  
Sierpiński Gasket ◽  
Average Probability ◽  
International Audience

Combinatorics International audience Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage n is a non-negative integer. For any given vertex x of SG(n), we derive rigorously the probability distribution of the degree j ∈{1,2,3,4} at the vertex and its value in the infinite n limit. Adding up such probabilities of all the vertices divided by the number of vertices, we obtain the average probability distribution of the degree j. The corresponding limiting distribution φj gives the average probability that a vertex is connected by 1, 2, 3 or 4 bond(s) among all the spanning tree configurations. They are rational numbers given as φ1=10957/40464, φ2=6626035/13636368, φ3=2943139/13636368, φ4=124895/4545456.

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