Andrew Rechnitzer

Lattice animals and heaps of dimers

The general quest ofthis paper is the search f or new classes ofsquare lattice animals that are both large and exactly enumerable. The starting point is a bijection between a subclass ofanimals, called directed animals, and certain heaps of dimers, called pyramids, which was described by Viennot more than 10 years ago. The generating function for directed animals had been known since 1982, but Viennot’s bijection suggested a new approach that greatly simpli7ed its derivation. We de7ne here two natural classes ofheaps that are supersets ofpyramids and are in bijection with certain classes ofanimals, and we enumerate them exactly. The 7rst class has an algebraic generating function and growth constant 3:5 (meaning that the number of n-celled animals grows like 3:5 n ), while the other has a transcendental non-holonomic generating function and growth constant 3:58 ::: :The generating function for directed animals is algebraic, and has growth constant 3. Hence both these new classes are exponentially larger. We obtain similar results for triangular lattice animals. c 2002 Elsevier Science B.V. All rights reserved.

Two non-holonomic lattice walks in the quarter plane

We present two classes of random walks restricted to the quarter plane with non-holonomic generating functions. The non-holonomicity is established using the iterated kernel method, a variant of the kernel method. This adds evidence to a recent conjecture on combinatorial properties of walks with holonomic generating functions [M. Mishna, Classifying lattice walks in the quarter plane, J. Combin. Theory Ser. A 116 (2009) 460-477]. The method also yields an asymptotic expression for the number of walks of length n.

On the Stanley--Wilf limit of 4231-avoiding permutations and a conjecture of Arratia

We show that the Stanley-Wilf limit for the class of 4231-avoiding permutations is at least by 9.47. This bound shows that this class has the largest such limit among all classes of permutations avoiding a single permutation of length 4 and refutes the conjecture that the Stanley-Wilf limit of a class of permutations avoiding a single permutation of length k cannot exceed (k-1)^2. The result is established by constructing a sequence of finite automata that accept subclasses of the class of 4231-avoiding permutations and analysing their transition matrices.

A directed walk model of a long chain polymer in a slit with attractive walls

We present the exact solutions of various directed walk models of polymers confined to a slit and interacting with the walls of the slit via an attractive potential. We consider three geometric constraints on the ends of the polymer and concentrate on the long chain limit. Apart from the general interest in the effect of geometrical confinement, this can be viewed as a two-dimensional model of steric stabilization and sensitized flocculation of colloidal dispersions. We demonstrate that the large width limit admits a phase diagram that is markedly different from the one found in a half-plane geometry, even when the polymer is constrained to be fixed at both ends on one wall. We are not able to find a closed form solution for the free energy for finite width, at all values of the interaction parameters, but we can calculate the asymptotic behaviour for large widths everywhere in the phase plane. This allows us to find the force between the walls induced by the polymer and hence the regions of the plane where either steric stabilization or sensitized flocculation would occur.

The site-perimeter of bargraphs

The site-perimeter enumeration of polyominoes that are both column- and row-convex is a well understood problem that always yields algebraic generating functions. Counting more general families of polyominoes is a far more difficult problem. Here we enumerate (by their site-perimeter) the simplest family of polyominoes that are not fully convex-bargraphs. The generating function we obtain is of a type that, to our knowledge, has never been encountered so far in the combinatorics literature: a q-series into which an algebraic series has been substituted.

Layering transitions for adsorbing polymers in poor solvents

An infinite hierarchy of layering transitions exists for model polymers in solution under poor solvent or low temperatures and near an attractive surface. A flat histogram stochastic growth algorithm known as FlatPERM has been used on a self- and surface interacting self-avoiding walk model for lengths up to 256. The associated phases exist as stable equilibria for large though not infinite length polymers and break the conjectured Surface Attached Globule phase into a series of phases where a polymer exists in specified layer close to a surface. We provide a scaling theory for these phases and the first-order transitions between them.

Stretching of a chain polymer adsorbed at a surface

In this paper we present simulations of a surface-adsorbed polymer subject to an elongation force. The polymer is modelled by a self-avoiding walk on a regular lattice. It is confined to a half-space by an adsorbing surface with attractions for every vertex of the walk visiting the surface, and the last vertex is pulled perpendicular to the surface by a force. Using the recently proposed flatPERM algorithm, we calculate the phase diagram for a vast range of temperatures and forces. The strength of this algorithm is that it computes the complete density of states from one single simulation. We simulate systems of sizes up to 256 steps.

Haruspicy and anisotropic generating functions

Guttmann and Enting [Phys. Rev. Lett. 76 (1996) 344-347] proposed the examination of anisotropic generating functions as a test of the solvability of models of bond animals. In this article we describe a technique for examining some properties of anisotropic generating functions. For a wide range of solved and unsolved families of bond animals, we show that the coefficients of yn is rational, the degree of its numerator is at most that of its denominator, and the denominator is a product of cyclotomic polynomials. Further we are able to find a multiplicative upper bound for these denominators which, by comparison with numerical studies [Jensen, personal communication; Jensen and Guttmann, personal communication], appears to be very tight. These facts can be used to greatly reduce the amount of computation required in generating series expansions. They also have strong and negative implications for the solvability of these problems.

Pulling absorbing and collapsing polymers from a surface

A self-interacting polymer with one end attached to a sticky surface has been studied by means of a flat-histogram stochastic growth algorithm known as FlatPERM. We examined the four-dimensional parameter space of the number of monomers (up to 91), self-attraction, surface-attraction and pulling force applied to one end of the polymer. Using this powerful algorithm the complete parameter space of interactions and pulling force has been considered. Recently it has been conjectured that a hierarchy of states appears at low-temperature/poor solvent conditions where a polymer exists in a finite number of layers close to a surface. We find re-entrant behaviour from the stretched phase into these layering phases when an appropriate force is applied to the polymer. Of interest is that the existence, and extent, of this re-entrant phase can be controlled not only by the force, but also by the ratio of surface-attraction to self-attraction. We also find that, contrary to what may be expected, the polymer desorbs from the surface when a sufficiently strong critical force is applied and does not transcend through either a series of de-layering transitions or monomer-by-monomer transitions. We discuss the problem mainly from the point of view of the stress ensemble. However, we make some comparisons with the strain ensemble, showing the broad agreement between the two ensembles while pointing out subtle differences.

Random copolymers and the Morita approximation: polymer adsorption and polymer localization

We analyse directed walk models of random copolymer adsorption and localization. Ideally we would like t os olve thequenched problem, but it appears to be intractable even for simple directed models. The annealed approximation is solvable, but is inadequate in the strong interaction regime fo rt he adsorption problem and gives a qualitatively incorrect phase diagram for the localization problem. In this paper, we treat these directed models using an approximation suggested by Morita (1964 J. Math. Phys. 5 1401–5) in which the proportion of each comonomer is fixed. We find that the Morita approximation leads to behaviour that is closer to that of the quenched average model and this is particularly interesting in the localization problem where the phase diagram is (a tl east qualitatively) very similar to that of the quenched average problem. We also show that the phase boundaries in the Morita approximation are bounds on the locations of the phase boundaries of the quenched model.