Heaps Of Pieces Research Articles

From steady-state TASEP model with open boundaries to 1D Ising model at negative fugacity

We expose a series of exact mappings between particular cases of four statistical physics models: (i) equilibrium 1D lattice gas with nearest-neighbor repulsion, (ii) (1 + 1)D combinatorial heap of pieces, (iii) directed random walks on a half-plane, and (iv) 1D totally asymmetric simple exclusion process (TASEP). In particular, we show that generating function of a 1D steady-state TASEP with open boundaries can be interpreted as a quotient of partition functions of 1D hard-core lattice gases with one adsorbing lattice site and negative fugacity. This result is based on the combination of a representation of a steady-state TASEP configurations in terms of (1 + 1)D heaps of pieces (HP) and a theorem of X Viennot which projects the partition function of (1 + 1)D HP onto that of a single layer of pieces, which in this case is a 1D hard-core lattice gas.

Generalised Chromatic Polynomials of Graphs from Heaps of Pieces

Generalised Chromatic Polynomials of Graphs from Heaps of Pieces

Toric Heaps, Cyclic Reducibility, and Conjugacy in Coxeter Groups

In 1986, G.X. Viennot introduced the theory of heaps of pieces as a visualization of Cartier and Foata’s “partially commutative monoids”. These are essentially labeled posets satisfying a few additional properties, and one natural setting where they arise is as models of reduced words in Coxeter groups. In this paper, we introduce a cyclic version of a heap, which loosely speaking, can be thought of as taking a heap and wrapping it into a cylinder. We call this object a toric heap, because we formalize it as a labeled toric poset, which is a cyclic version of an ordinary poset. Defining the category of toric heaps leads to the notion of certain morphisms such as toric extensions. We study toric heaps in Coxeter theory, because a cyclic shift of a reduced word is simply a conjugate by an initial or terminal generator. As such, we formalize and study a framework that we call cyclic reducibility in Coxeter theory, which is closely related to conjugacy. We introduce what it means for elements to be torically reduced, which is a stronger condition than simply being cyclically reduced. Along the way, we encounter a new class of elements that we call torically fully commutative (TFC), which are those that have a unique cyclic commutativity class, and comprise a strictly bigger class than the cyclically fully commutative (CFC) elements. We prove several cyclic analogues of results on fully commutative (FC) elements due to Stembridge. We conclude with how this framework fits into recent work in Coxeter groups, and we correct a minor flaw in a few recently published theorems.

Combinatorial Hopf algebras from PROs

We introduce a general construction that takes as input a so-called stiff PRO and that outputs a Hopf algebra. Stiff PROs are particular PROs that can be described by generators and relations with precise conditions. Our construction generalizes the classical construction from operads to Hopf algebras of van der Laan. We study some of its properties and review some examples of application. We get in particular Hopf algebras on heaps of pieces and retrieve some deformed versions of the noncommutative Fa\`a di Bruno algebra introduced by Foissy.

Uniform and Bernoulli measures on the boundary of trace monoids

Trace monoids and heaps of pieces appear in various contexts in combinatorics. They also constitute a model used in computer science to describe the executions of asynchronous systems. The design of a natural probabilistic layer on top of the model has been a long standing challenge. The difficulty comes from the presence of commuting pieces and from the absence of a global clock. In this paper, we introduce and study the class of Bernoulli probability measures that we claim to be the simplest adequate probability measures on infinite traces. For this, we strongly rely on the theory of trace combinatorics with the Möbius polynomial in the key role. These new measures provide a theoretical foundation for the probabilistic study of concurrent systems.

Some Structural Properties of Acyclic Heaps of Pieces

We introduce the notions of boundary vertex, linear equivalence, and effective boundary vertex in the context of Viennot’s heaps of pieces. We prove that in the heap of a fully commutative element in a star reducible Coxeter group, every boundary vertex is linearly equivalent to an effective boundary vertex.

Average site perimeter of directed animals on the two-dimensional lattices

We introduce new combinatorial (bijective) methods that enable us to compute the average value of three parameters of directed animals of a given area, including the site perimeter. Our results cover directed animals of any one-line source on the square lattice and its bounded variants, and we give counterparts for most of them in the triangular lattices. We thus prove conjectures by Conway and Le Borgne. The techniques used are based on Viennot’s correspondence between directed animals and heaps of pieces (or elements of a partially commutative monoid).

Computing the average parallelism in trace monoids

The height of a trace is the height of the corresponding heap of pieces in Viennot's representation, or equivalently the number of factors in its Cartier–Foata decomposition. Let h( t) and | t| stand respectively for the height and the length of a trace t. We prove that the bivariate commutative series ∑ t x h( t) y | t| is rational, and we give a finite representation of it. We use the rationality to obtain precise information on the asymptotics of the number of traces of a given height or length. Then, we study the average height of a trace for various probability distributions on traces. For the uniform probability distribution on traces of the same length (resp. of the same height), the asymptotic average height (resp. length) exists and is an algebraic number. To illustrate our results and methods, we consider a couple of examples: the free commutative monoid and the trace monoid whose independence graph is the ladder graph.

On rank functions for heaps

Motivated by work of Stembridge, we study rank functions for Viennot's heaps of pieces. We produce a simple and sufficient criterion for a heap to be a ranked poset and apply the results to the heaps arising from fully commutative words in Coxeter groups.

On the average Cartier-Foata height of traces

The height of a trace is the height of the corresponding heap of pieces in X.G. Viennot's formalism, or equivalently the number of components in its Cartier-Foata decomposition. Let h(t) and ∣t∣ be, respectively, the height and the length of a trace t. We prove that the bivariate commutative series ∑tx ∣t∣yh(t) is rational, and we give a finite representation of it. As a by-product we obtain that the asymptotic average height of the traces of a given length is rational and explicitly computable. We show how to exploit the symmetries in the dependence graph to obtain representations of reduced dimensions of the series. For highly symmetric trace monoids, the computations may become very effective. To illustrate this point, we consider the family of trace monoids whose dependence graphs are triangular graphs, i.e., line graphs of complete graphs. We study the combinatorics of this case in details.

Bijections for Directed Animals on Infinite Families of Lattices

Bousquet-Melou and Conwayin [1] found algebraic equations for the area generating function of directed animals on an infinite family of regular, non-planar, two-dimensional lattices by using equivalences with hard particle models. We give in this paper a bijective proof of their results which is a generalization of Viennot's heaps of pieces [10,12]. Based on this proof we get exacte numeration formulas for the number of configurations with area n which could not be deduced directly from the algebraic equation. Moreover, we give an extension of these results to another infinite family of lattices.

Lyndon heaps: An analogue of Lyndon words in free partially commutative monoids

Given a free partially commutative monoid, we interpret equivalence classes of words (also called ‘traces’) in terms of Viennot's heaps of pieces. We study the notion of conjugacy applied to this context. Each heap has a unique representative in the set of ‘standard words’ which are lexicographically ordered. Then, by definition, Lyndon heaps are the minimal elements of the conjugacy classes of heaps. Most combinatorial properties of Lyndon words have a counterpart in the theory of Lyndon heaps. For instance, a ‘standard factorization’ exists, the set of Lyndon heaps provides a complete factorization of the monoid, etc.