scholarly journals Cumulants of the q-semicircular law, Tutte polynomials, and heaps

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Matthieu Josuat-Vergès
Keyword(s):  
Tutte Polynomial ◽  
Combinatorial Interpretation ◽  
Poisson Law ◽  
Semicircular Law ◽  
Combinatorial Model ◽  
Tutte Polynomials ◽  
International Audience ◽  
Nonnegative Coefficients

International audience The q-semicircular law as introduced by Bożejko and Speicher interpolates between the Gaussian law and the semicircular law, and its moments have a combinatorial interpretation in terms of matchings and crossings. We prove that the cumulants of this law are, up to some factor, polynomials in q with nonnegative coefficients. This is done by showing that they are obtained by an enumeration of connected matchings, weighted by the evaluation at (1,q) of a Tutte polynomial. The two particular cases q=0 and q=2 have also alternative proofs, related with the fact that these particular evaluation of the Tutte polynomials count some orientations on graphs. Our methods also give a combinatorial model for the cumulants of the free Poisson law. La loi q-semicirculaire introduite par Bożejko et Speicher interpole entre la loi gaussienne et la loi semi-circulaire, et ses moments ont une interprétation combinatoire en termes de couplages et croisements. Nous prouvons que les cumulants de cette loi sont, à un facteur près, des polynômes en q à coefficients positifs. La méthode consiste à obtenir ces cumulants par une énumération de couplages connexes, pondérés par l’évaluation en (1,q) d'un polynôme de Tutte. Les cas particuliers q=0 et q=2 ont une preuve alternative, reliè au fait que des évaluations particulières du polynôme de Tutte comptent certaines orientations de graphes. Nos méthodes donnent aussi un modèle combinatoire aux cumulants de la loi de Poisson libre.

scholarly journals Arithmetic matroids and Tutte polynomials

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Michele D'Adderio ◽  
Luca Moci
Keyword(s):  
Abelian Group ◽  
Tutte Polynomial ◽  
Finitely Generated ◽  
Combinatorial Interpretation ◽  
Tutte Polynomials ◽  
International Audience

International audience We introduce the notion of arithmetic matroid, whose main example is provided by a list of elements in a finitely generated abelian group. We study the representability of its dual, and, guided by the geometry of toric arrangements, we give a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's formula. Nous introduisons la notion de matroï de arithmètique, dont le principal exemple est donnè par une liste d'élèments dans un groupe abèlien fini. Nous ètudions la reprèsentabilitè de son dual, et, guidè par la gèomètrie des arrangements toriques, nous donnons une interprètation combinatoire du polynôme de Tutte arithmètique associèe, ce qui peut être vu comme une gènèralisation de la formule de Crapo.

scholarly journals Cumulants of the q-semicircular Law, Tutte Polynomials, and Heaps

2013 ◽  
Vol 65 (4) ◽  
pp. 863-878 ◽  
Author(s):  
Matthieu Josuat Vergès
Keyword(s):  
General Formula ◽  
Hypergeometric Series ◽  
Symmetric Functions ◽  
Tutte Polynomial ◽  
Geometric Interpretation ◽  
Combinatorial Interpretation ◽  
Free Cumulants ◽  
Combinatorial Properties ◽  
Semicircular Law ◽  
Tutte Polynomials

AbstractThe q-semicircular distribution is a probability law that interpolates between the Gaussian law and the semicircular law. There is a combinatorial interpretation of itsmoments in terms ofmatchings, where q follows the number of crossings, whereas for the free cumulants one has to restrict the enumeration to connected matchings. The purpose of this article is to describe combinatorial properties of the classical cumulants. We show that like the free cumulants, they are obtained by an enumeration of connected matchings, the weight being now an evaluation of the Tutte polynomial of a so-called crossing graph. The case q = 0 of these cumulants was studied by Lassalle using symmetric functions and hypergeometric series. We show that the underlying combinatorics is explained through the theory of heaps, which is Viennot's geometric interpretation of the Cartier–Foata monoid. This method also gives a general formula for the cumulants in terms of free cumulants.

scholarly journals Zonotopes, toric arrangements, and generalized Tutte polynomials

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Luca Moci
Keyword(s):  
Characteristic Polynomial ◽  
Hilbert Series ◽  
Hyperplane Arrangement ◽  
Tutte Polynomial ◽  
Integral Points ◽  
Poincaré Polynomial ◽  
Tutte Polynomials ◽  
International Audience ◽  
Toric Arrangement ◽  
Positive Coefficients

International audience We introduce a multiplicity Tutte polynomial $M(x,y)$, which generalizes the ordinary one and has applications to zonotopes and toric arrangements. We prove that $M(x,y)$ satisfies a deletion-restriction recurrence and has positive coefficients. The characteristic polynomial and the Poincaré polynomial of a toric arrangement are shown to be specializations of the associated polynomial $M(x,y)$, likewise the corresponding polynomials for a hyperplane arrangement are specializations of the ordinary Tutte polynomial. Furthermore, $M(1,y)$ is the Hilbert series of the related discrete Dahmen-Micchelli space, while $M(x,1)$ computes the volume and the number of integral points of the associated zonotope. On introduit un polynôme de Tutte avec multiplicité $M(x, y)$, qui généralise le polynôme de Tutte ordinaire et a des applications aux zonotopes et aux arrangements toriques. Nous prouvons que $M(x, y)$ satisfait une récurrence de "deletion-restriction'' et a des coefficients positifs. Le polynôme caractéristique et le polynôme de Poincaré d'un arrangement torique sont des spécialisations du polynôme associé $M(x, y)$, de même que les polynômes correspondants pour un arrangement d'hyperplans sont des spécialisations du polynôme de Tutte ordinaire. En outre, $M(1, y)$ est la série de Hilbert de l'espace discret de Dahmen-Micchelli associé, et $M(x, 1)$ calcule le volume et le nombre de points entiers du zonotope associé.

scholarly journals The arithmetic Tutte polynomials of the classical root systems

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Federico Ardila ◽  
Federico Castillo ◽  
Michael Henley
Keyword(s):  
Hyperplane Arrangement ◽  
Tutte Polynomial ◽  
Root Systems ◽  
Field Method ◽  
Topological Invariants ◽  
Signed Graphs ◽  
Tutte Polynomials ◽  
International Audience

International audience Many combinatorial and topological invariants of a hyperplane arrangement can be computed in terms of its Tutte polynomial. Similarly, many invariants of a hypertoric arrangement can be computed in terms of its <i>arithmetic</i> Tutte polynomial. We compute the arithmetic Tutte polynomials of the classical root systems $A_n, B_n, C_n$, and $D_n$ with respect to their integer, root, and weight lattices. We do it in two ways: by introducing a \emphfinite field method for arithmetic Tutte polynomials, and by enumerating signed graphs with respect to six parameters.

scholarly journals Computing Tutte Polynomials

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Michael Monagan
Keyword(s):  
Recent Result ◽  
Ad Hoc ◽  
Tutte Polynomial ◽  
Nous Pouvons ◽  
Sparse Graphs ◽  
Vertex Ordering ◽  
Tutte Polynomials ◽  
International Audience

International audience We present a new edge selection heuristic and vertex ordering heuristic that together enable one to compute the Tutte polynomial of much larger sparse graphs than was previously doable. As a specific example, we are able to compute the Tutte polynomial of the truncated icosahedron graph using our Maple implementation in under 4 minutes on a single CPU. This compares with a recent result of Haggard, Pearce and Royle whose special purpose C++ software took one week on 150 computers. Nous présentons deux nouvelles heuristiques pour le calcul du polynôme de Tutte de graphes de faible densité, basées sur les principes de sélection d'arêtes et d'arrangement linéaire de sommets, et qui permettent de traiter des graphes de bien plus grande tailles que les méthodes existantes. Par exemple, en utilisant notre implémentation en Maple, nous pouvons calculer le polynôme de Tutte de l'isocahédron tronqué en moins de 4 minutes sur un ordinateur à processeur unique, alors qu'un programme ad-hoc récent de Haggard, Pearce et Royle, utilisant 150 ordinateurs, a nécessité une semaine de calcul pour obtenir le même résultat.

scholarly journals A lattice point counting generalisation of the Tutte polynomial

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Amanda Cameron ◽  
Alex Fink
Keyword(s):  
Lattice Point ◽  
Tutte Polynomial ◽  
Minkowski Sum ◽  
Point Counting ◽  
Combinatorial Interpretation ◽  
Point Counts ◽  
Standard Simplex ◽  
International Audience

International audience The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion- contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. This polynomial is constructed using lattice point counts in the Minkowski sum of the base polytope of a polymatroid and scaled copies of the standard simplex. We also show that, in the matroid case, our polynomial has coefficients of alternating sign, with a combinatorial interpretation closely tied to the Dawson partition.

scholarly journals The Tutte polynomial of symmetric hyperplane arrangements

2020 ◽  
Vol 29 (03) ◽  
pp. 2050004
Author(s):  
Hery Randriamaro
Keyword(s):  
Hyperplane Arrangement ◽  
Dual Graph ◽  
Tutte Polynomial ◽  
Hyperplane Arrangements ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Tutte Polynomials ◽  
One Year ◽  
The One ◽  
Definition Of

The Tutte polynomial is originally a bivariate polynomial which enumerates the colorings of a graph and of its dual graph. Ardila extended in 2007 the definition of the Tutte polynomial on the real hyperplane arrangements. He particularly computed the Tutte polynomials of the hyperplane arrangements associated to the classical Weyl groups. Those associated to the exceptional Weyl groups were computed by De Concini and Procesi one year later. This paper has two objectives: On the one side, we extend the Tutte polynomial computing to the complex hyperplane arrangements. On the other side, we introduce a wider class of hyperplane arrangements which is that of the symmetric hyperplane arrangements. Computing the Tutte polynomial of a symmetric hyperplane arrangement permits us to deduce the Tutte polynomials of some hyperplane arrangements, particularly of those associated to the imprimitive reflection groups.

scholarly journals A study about the Tutte polynomials of benzenoid chains

2017 ◽  
Vol 5 (1) ◽  
pp. 28-32
Author(s):  
Abdulgani Sahin
Keyword(s):  
Tutte Polynomial ◽  
Signed Graphs ◽  
Tutte Polynomials

Abstract The Tutte polynomials for signed graphs were introduced by Kauffman. In 2012, Fath-Tabar, Gholam-Rezaeı and Ashrafı presented a formula for computing Tutte polynomial of a benzenoid chain. From this point on, we have also calculated the Tutte polynomials of signed graphs of benzenoid chains in this study.

scholarly journals Combinatorics of k-shapes and Genocchi numbers

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Florent Hivert ◽  
Olivier Mallet
Keyword(s):  
Symmetric Functions ◽  
New Model ◽  
Work In Progress ◽  
Genocchi Numbers ◽  
Combinatorial Model ◽  
International Audience

International audience In this paper we present a work in progress on a conjectural new combinatorial model for the Genocchi numbers. This new model called irreducible k-shapes has a strong algebraic background in the theory of symmetric functions and leads to seemingly new features on the theory of Genocchi numbers. In particular, the natural q-analogue coming from the degree of symmetric functions seems to be unknown so far. Dans cet article, nous présentons un travail en cours sur un nouveau modèle combinatoire conjectural pour les nombres de Genocchi. Ce nouveau modèle est celui des k-formes irréductibles, qui repose sur de solides bases algébriques en lien avec la théorie des fonctions symétriques et qui conduit à des aspects apparemment nouveaux de la théorie des nombres de Genocchi. En particulier, le q-analogue naturel venant du degré des fonctions symétriques semble inconnu jusqu'ici.

scholarly journals The Cube Recurrence

10.37236/1826 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Gabriel D. Carroll ◽  
David Speyer
Keyword(s):  
Recurrence Relation ◽  
Planar Graphs ◽  
Perfect Matchings ◽  
Laurent Polynomials ◽  
Combinatorial Interpretation ◽  
Combinatorial Model

We construct a combinatorial model that is described by the cube recurrence, a quadratic recurrence relation introduced by Propp, which generates families of Laurent polynomials indexed by points in ${\Bbb Z}^3$. In the process, we prove several conjectures of Propp and of Fomin and Zelevinsky about the structure of these polynomials, and we obtain a combinatorial interpretation for the terms of Gale-Robinson sequences, including the Somos-6 and Somos-7 sequences. We also indicate how the model might be used to obtain some interesting results about perfect matchings of certain bipartite planar graphs.

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