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 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 Graph polynomials and symmetries

2019 ◽  
Vol 18 (09) ◽  
pp. 1950172 ◽  
Author(s):  
Nafaa Chbili
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Characteristic Polynomial ◽  
Prime Order ◽  
Tutte Polynomial ◽  
Quotient Graph ◽  
Graph Polynomials ◽  
Tutte Polynomials

In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of this paper is to extend this study to other graph polynomials. In particular, we prove that if a graph [Formula: see text] has a symmetry of prime order [Formula: see text], then its characteristic polynomial, with coefficients in the finite field [Formula: see text], is determined by the characteristic polynomial of its quotient graph [Formula: see text]. Similar results are also proved for some generalization of the Tutte polynomial.

scholarly journals Hierarchical Zonotopal Power Ideals

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Matthias Lenz
Keyword(s):  
Characteristic Polynomial ◽  
Hilbert Series ◽  
Finite Sequence ◽  
Vector Spaces ◽  
Geometric Structures ◽  
Linear Forms ◽  
International Audience ◽  
Powers Of Linear Forms ◽  
Matroid Structure

International audience Zonotopal algebra deals with ideals and vector spaces of polynomials that are related to several combinatorial and geometric structures defined by a finite sequence of vectors. Given such a sequence $X$, an integer $k \geq -1$ and an upper set in the lattice of flats of the matroid defined by $X$, we define and study the associated $\textit{hierarchical zonotopal power ideal}$. This ideal is generated by powers of linear forms. Its Hilbert series depends only on the matroid structure of $X$. It is related to various other matroid invariants, $\textit{e. g.}$ the shelling polynomial and the characteristic polynomial. This work unifies and generalizes results by Ardila-Postnikov on power ideals and by Holtz-Ron and Holtz-Ron-Xu on (hierarchical) zonotopal algebra. We also generalize a result on zonotopal Cox modules due to Sturmfels-Xu. La théorie de l'algèbre "zonotopique'' s'occupe d'idéaux et d'espaces vectoriels de polynômes qui ont un rapport avec plusieurs structures combinatoires et géométriques définies par des suites finies de vecteurs. Étant donné une telle suite $X$, un nombre entier $k \geq -1$ et un ensemble supérieur dans le treillis des plans du matroïde défini par $X$, nous définissons et étudions l'$\textit{idéal hiérarchique zonotopique}$, engendré par des puissances de formes linéaires. Sa série de Hilbert dépend seulement de la structure matroïdale de $X$. Il existe des relations avec d'autres invariants de matroïdes, tels que le polynôme d'épluchage et le polynôme caractéristique. Ce travail unifie et généralise des résultats d'Ardila-Postnikov sur les idéaux de puissances et de Holtz-Ron et Holtz-Ron-Xu sur l'algèbre zonotopique (hiérarchique). Nous généralisons aussi un résultat sur les modules de Cox zonotopiques, dû à Sturmfels-Xu.

scholarly journals Bruhat order, rationally smooth Schubert varieties, and hyperplane arrangements

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Suho Oh ◽  
Hwanchul Yoo
Keyword(s):  
Hyperplane Arrangement ◽  
Schubert Variety ◽  
Bruhat Order ◽  
Hyperplane Arrangements ◽  
Schubert Varieties ◽  
Flag Manifolds ◽  
Poincaré Polynomial ◽  
Generalized Flag Manifolds ◽  
Nous Montrons ◽  
International Audience

International audience We link Schubert varieties in the generalized flag manifolds with hyperplane arrangements. For an element of a Weyl group, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is rationally smooth. Nous relions des variétés de Schubert dans le variété flag généralisée avec des arrangements des hyperplans. Pour un élément dún groupe de Weyl, nous construisons un certain arrangement graphique des hyperplans. Nous montrons que la fonction génératrice pour les régions de cet arrangement coincide avec le polynome de Poincaré de la variété de Schubert correspondante si et seulement si la variété de Schubert est rationnellement lisse.

scholarly journals Hyperplane Arrangements and Diagonal Harmonics

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Drew Armstrong
Keyword(s):  
Symmetric Functions ◽  
Hilbert Series ◽  
Hyperplane Arrangement ◽  
Type A ◽  
Hyperplane Arrangements ◽  
Combinatorial Interpretation ◽  
Nabla Operator ◽  
Affine Weyl Group ◽  
Diagonal Harmonics ◽  
International Audience

International audience In 2003, Haglund's bounce statistic gave the first combinatorial interpretation of the q,t-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type A. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement — which we call the Ish arrangement. We prove that our statistics are equivalent to the area' and bounce statistics of Haglund and Loehr. In this setting, we observe that bounce is naturally expressed as a statistic on the root lattice. We extend our statistics in two directions: to "extended'' Shi arrangements and to the bounded chambers of these arrangements. This leads to a (conjectural) combinatorial interpretation for all integral powers of the Bergeron-Garsia nabla operator applied to elementary symmetric functions. En 2003, la statistique bounce de Haglund a donné la première interprétation combinatoire de la somme des nombres q,t-Catalan et de la série de Hilbert des harmoniques diagonaux. Dans cet article nous proposons une nouvelle interprétation combinatoire à partir du groupe de Weyl affine de type A. En particulier, nous définissons deux statistiques sur les permutations affines; l'une à partir de l'arrangement d'hyperplans Shi, et l'autre à partir d'un nouvel arrangement — que nous appelons l'arrangement Ish. Nous prouvons que nos statistiques sont équivalentes aux statistiques area' et bounce de Haglund et Loehr. Dans ce contexte, nous observons que bounce s'exprime naturellement comme une statistique sur le réseau des racines. Nous prolongeons nos statistiques dans deux directions: arrangements Shi "étendus'', et chambres bornées associées. Cela conduit à une interprétation (conjecturale) combinatoire pour toutes les puissances entières de l'opérateur nabla de Bergeron-Garsia appliqué aux fonctions symétriques élémentaires.

The Tutte polynomial of ideal arrangements

2020 ◽  
Vol 12 (02) ◽  
pp. 2050017
Author(s):  
Hery Randriamaro
Keyword(s):  
Finite Field ◽  
Hyperplane Arrangement ◽  
Dual Graph ◽  
Tutte Polynomial ◽  
Root Systems ◽  
Field Method ◽  
Hyperplane Arrangements ◽  
Acyclic Orientations ◽  
The Matrix ◽  
Tutte Polynomials

The Tutte polynomial was originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of acyclic orientations. In 2007, Ardila extended the notion of Tutte polynomial to hyperplane arrangements, and computed the Tutte polynomials of the classical root systems for certain prime powers of the first variable at the same time. In this paper, we compute the Tutte polynomial of ideal arrangements. These arrangements were introduced in 2006 by Sommers and Tymoczko, and are defined for ideals of root systems. For the ideals of classical root systems, we bring a slight improvement of the finite field method by showing that it can applied on any finite field whose cardinality is not a minor of the matrix associated to the studied hyperplane arrangement. Computing the minor set associated to an ideal of classical root systems particularly permits us to deduce the Tutte polynomials of the classical root systems. For the ideals of the exceptional root systems of type [Formula: see text], [Formula: see text], and [Formula: see text], we use the formula of Crapo.

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 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.

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