scholarly journals Improved Bounds for the Number of Forests and Acyclic Orientations in the Square Lattice

10.37236/1697 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
N. Calkin ◽  
C. Merino ◽  
S. Noble ◽  
M. Noy
Keyword(s):  
Tutte Polynomial ◽  
Square Lattice ◽  
Transfer Matrices ◽  
Counting Problems ◽  
Acyclic Orientations

In a recent paper Merino and Welsh (1999) studied several counting problems on the square lattice $L_n$. There the authors gave the following bounds for the asymptotics of $f(n)$, the number of forests of $L_n$, and $\alpha(n)$, the number of acyclic orientations of $L_n$: $$3.209912 \le \lim_{n\to\infty} f(n)^{1/n^2} \le 3.84161$$ and $$22/7 \le \lim_{n\to\infty} \alpha(n)^{1/n^2} \le 3.70925.$$ In this paper we improve these bounds as follows: $$3.64497 \le \lim_{n\to\infty} f(n)^{1/n^2} \le 3.74101$$ and $$3.41358 \le \lim_{n\to\infty} \alpha(n)^{1/n^2} \le 3.55449.$$ We obtain this by developing a method for computing the Tutte polynomial of the square lattice and other related graphs based on transfer matrices.

Series expansions from corner transfer matrices: The square lattice Ising model

1979 ◽  
Vol 21 (2) ◽  
pp. 103-123 ◽  
Author(s):  
R. J. Baxter ◽  
I. G. Enting
Keyword(s):  
Ising Model ◽  
Square Lattice ◽  
Series Expansions ◽  
Transfer Matrices ◽  
Corner Transfer Matrices

General Square Lattice Vertex Models

2019 ◽  
pp. 430-453
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Square Lattice ◽  
Vertex Model ◽  
Transfer Matrices ◽  
Spectral Parameters ◽  
Vertex Models ◽  
Integrable Quantum ◽  
Special Cases ◽  
Mechanical Models ◽  
The One ◽  
Ice Model

Vertex models more general than the ice model are possible and often have physical applications. The square lattice admits the general sixteen-vertex model of which the special cases, the eight- and the six-vertex model, are the most relevant and physically interesting, in particular through their connection to the one-dimensional integrable quantum mechanical models and the Bethe ansatz. This chapter introduces power- ful tools to examine vertex models, including the R- and L-matrices to encode the Boltzmann vertex weights and the monodromy and transfer matrices, which encode the integrability of the vertex models (i.e. that transfer matrices of different spectral parameters commute). This integrability is ultimately expressed in the Yang–Baxter relations.

scholarly journals On some Tutte polynomial sequences in the square lattice

2012 ◽  
Vol 102 (2) ◽  
pp. 436-453 ◽  
Author(s):  
Arun P. Mani
Keyword(s):  
Tutte Polynomial ◽  
Square Lattice ◽  
Polynomial Sequences

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 Cylindric Hecke Characters and Gromov–Witten Invariants via the Asymmetric Six-Vertex Model

2020 ◽  
Author(s):  
Christian Korff
Keyword(s):  
Hecke Algebras ◽  
Square Lattice ◽  
Vertex Operators ◽  
Vertex Model ◽  
Transfer Matrices ◽  
Genus Zero ◽  
Planar Lattice ◽  
Infinite Dimensional ◽  
Structure Constants ◽  
Restriction Functor

AbstractWe construct a family of infinite-dimensional positive sub-coalgebras within the Grothendieck ring of Hecke algebras, when viewed as a Hopf algebra with respect to the induction and restriction functor. These sub-coalgebras have as structure constants the 3-point genus zero Gromov–Witten invariants of Grassmannians and are spanned by what we call cylindric Hecke characters, a particular set of virtual characters for whose computation we give several explicit combinatorial formulae. One of these expressions is a generalisation of Ram’s formula for irreducible Hecke characters and uses cylindric broken rim hook tableaux. We show that the latter are in bijection with so-called ‘ice configurations’ on a cylindrical square lattice, which define the asymmetric six-vertex model in statistical mechanics. A key ingredient of our construction is an extension of the boson-fermion correspondence to Hecke algebras and employing the latter we find new expressions for Jing’s vertex operators of Hall–Littlewood functions in terms of the six-vertex transfer matrices on the infinite planar lattice.

scholarly journals The Active Bijection between Regions and Simplices in Supersolvable Arrangements of Hyperplanes

10.37236/1887 ◽  
2006 ◽  
Vol 11 (2) ◽  
Author(s):  
Emeric Gioan ◽  
Michel Las Vergnas
Keyword(s):  
Tutte Polynomial ◽  
Oriented Matroids ◽  
Oriented Matroid ◽  
Natural Activity ◽  
Acyclic Orientations ◽  
Linear Orderings ◽  
Numerical Relation ◽  
Arrangements Of Hyperplanes ◽  
Broken Circuit

Comparing two expressions of the Tutte polynomial of an ordered oriented matroid yields a remarkable numerical relation between the numbers of reorientations and bases with given activities. A natural activity preserving reorientation-to-basis mapping compatible with this relation is described in a series of papers by the present authors. This mapping, equivalent to a bijection between regions and no broken circuit subsets, provides a bijective version of several enumerative results due to Stanley, Winder, Zaslavsky, and Las Vergnas, expressing the number of acyclic orientations in graphs, or the number of regions in real arrangements of hyperplanes or pseudohyperplanes (i.e. oriented matroids), as evaluations of the Tutte polynomial. In the present paper, we consider in detail the supersolvable case – a notion introduced by Stanley – in the context of arrangements of hyperplanes. For linear orderings compatible with the supersolvable structure, special properties are available, yielding constructions significantly simpler than those in the general case. As an application, we completely carry out the computation of the active bijection for the Coxeter arrangements $A_n$ and $B_n$. It turns out that in both cases the active bijection is closely related to a classical bijection between permutations and increasing trees.

Forests, colorings and acyclic orientations of the square lattice

1999 ◽  
Vol 3 (2-4) ◽  
pp. 417-429 ◽  
Author(s):  
C. Merino ◽  
D. J. A. Welsh
Keyword(s):  
Square Lattice ◽  
Acyclic Orientations

scholarly journals The Tutte Polynomial of a Graph, Depth-first Search

10.37236/1267 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Ira M. Gessel ◽  
Bruce E. Sagan
Keyword(s):  
Spanning Trees ◽  
Tutte Polynomial ◽  
Simplicial Complexes ◽  
Parking Functions ◽  
Depth First Search ◽  
Acyclic Orientations

One of the most important numerical quantities that can be computed from a graph $G$ is the two-variable Tutte polynomial. Specializations of the Tutte polynomial count various objects associated with $G$, e.g., subgraphs, spanning trees, acyclic orientations, inversions and parking functions. We show that by partitioning certain simplicial complexes related to $G$ into intervals, one can provide combinatorial demonstrations of these results. One of the primary tools for providing such a partition is depth-first search.

scholarly journals The Incidence Hopf Algebra of Graphs

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Brandon Humpert ◽  
Jeremy L. Martin
Keyword(s):  
Hopf Algebra ◽  
General Formula ◽  
Tutte Polynomial ◽  
Simple Graphs ◽  
Acyclic Orientations ◽  
Graph Algebra ◽  
International Audience ◽  
New Formula

International audience The graph algebra is a commutative, cocommutative, graded, connected incidence Hopf algebra, whose basis elements correspond to finite simple graphs and whose Hopf product and coproduct admit simple combinatorial descriptions. We give a new formula for the antipode in the graph algebra in terms of acyclic orientations; our formula contains many fewer terms than Schmitt's more general formula for the antipode in an incidence Hopf algebra. Applications include several formulas (some old and some new) for evaluations of the Tutte polynomial. L'algèbre de graphes est une algèbre d'incidence de Hopf commutative, cocommutative, graduée, et connexe, dont les éléments de base correspondent à des graphes finis simples et dont le produit et coproduit de Hopf admettent une description combinatoire simple. Nous présentons une nouvelle formule de l'antipode dans l'algèbre de graphes utilisant les orientations acycliques; notre formule contient beaucoup moins de termes que la formule générale de Schmitt pour l'antipode dans une algèbre d'incidence de Hopf. Les applications incluent plusieurs formules (connues et inconnues) pour les évaluations du polynôme de Tutte.

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