deco polyominoes permutations and random generation abstract in this paper we introduce a class of polyominoes called deco polyominoes in bijection with the set of permutations of the first k integers we evaluate some typical parameters for this class of polyominoes and define a linear algorithm for randomly generating them moreover we define the polyominoes corresponding to the directed animals on the hexagonal and triangular lattices and examine these polyominoes deco class as far as deco polyominoes corresponding to animals on the hexagonal lattice are concerned we find a bijection with a class of permutations finally we give linear algorithms for randomly generating both hexagonal and triangular deco polyominoes

low density series expansions for directed percolation on square and triangular lattices greatly extended series have been derived for moments of the pair connectedness for bond and site percolation on the directed square and triangular lattices the length of the various series has been at least doubled to more than 110 100 terms for the square lattice bond site problem and more than 55 terms for the bond and site problems on the triangular lattice analysis of the series leads to very accurate estimates for the critical parameters and generally seems to rule out simple rational values for the critical exponents the values of the critical exponents for the average cluster size parallel and perpendicular connectedness lengths are estimated by d 2 277 69 4 k d 1 733 825 25 and d 1 096 844 14 respectively an improved estimate for the percolation probability exponent is obtained from the scaling relation d k c 2 d 0 276 49 4 in all cases the leading correction to scaling term is analytic

high order finite difference techniques for linear singular perturbation boundary value problems abstract three high order finite difference techniques are developed for the second order singularity perturbed linear boundary value problem in one dimension taylor series expansions and error conversions are used for the development of the techniques convergence and stability conditions of these techniques are proved and some numerical results are given

directed compact percolation near a damp wall mean length and mean number of wall contacts key aspects of the cluster distribution in the case of directed compact percolation near a damp wall are derived as functions of the bulk occupation probability pand the wall occupation probability pw the mean length of finite clusters and mean number of contacts with the wall are derived exactly and we find that both results involve elliptic integrals and further multiple sum functions of two variables despite the added complication of these multiple sumfunctions ouranalysisshowsthatthecriticalbehaviourissimilartothedry wallcasewherethesefunctionsdonotappear wederivethecriticalamplitudes as a function of pw

solving singularly perturbed problems by a weak form integral equation with exponential trial functions abstract the second order singularly perturbed problem is transformed to a singularly perturbed parabolic type partial differential equation by using a fictitious time technique then we use green s second identity to derive a boundary integral equation in terms of the adjoint trefftz test functions namely a weak form integral equation method wfiem it accompanying with the exponential trial functions which are designed to satisfy the boundary conditions automatically can provide very accurate numerical solutions of linear and nonlinear singularly perturbed problems for the latter problem the iterative procedure is convergent very fast

a holonomic systems approach to special functions identities abstract we observe that many special functions are solutions of so called holonomic systems bernstein s deep theory of holonomic systems is then invoked to show that any identity involving sums and integrals of products of these special functions can be verified in a finite number of steps this is partially substantiated by an algorithm that proves terminating hypergeometric series identities and that is given both in english and in maple

a new two variable generating function for convex self avoiding polygons the width of a self avoiding polygon on the square lattice is defined as the minimal horizontal or vertical distance between two of its parallel edges if the polygons are convex this distance is internal the perimeter generating functions for such convex polygons whose widths exceed a threshold can be given explicitly from these expressions a two variable width and perimeter generating function can be constructed the corresponding phase diagram shows two types of critical behaviour which meet at a tricritical point

convex polyominoes and heaps of segments builds a one to one correspondence between some polyominoes called parallelogram polyominoes and some heaps of segments then the author derives explicit expressions for the generating function of convex polyominoes according to their height width and area the author also enumerates two subsets of convex polyominoes namely the directed and convex polyominoes and the parallelogram polyominoes according to these three parameters generating functions of polyominoes having a fixed width are also studied

number of almost convex polygons on the square lattice the generating function for the number nc n of almost convex polygons on the square lattice with concavity index c 1 and perimeter n is derived rigorously the asymptotic behaviour of nc n for large n is determined and this result confirms a conjecture by enting et al 1992

the complexity of computing the number of strings of given length in context free languages abstract computing the number of strings of given length contained in a language is related to classical problems of combinatorics formal languages and computational complexity here we study the complexity of this problem in the case of context free languages it is shown that for unambiguous context free languages such a computation is easy and can be carried out by efficient parallel algorithms on the contrary for some context free languages of ambiguity degree two the problem becomes intractable these results are related to other classical subjects concerning counting problems exponential time recognizable languages and sparse sets

mean length of finite clusters in directed compact percolation near a damp wall the mean length of finite clusters is derived exactly for the case of directed compact percolation near a damp wall we find that the result involves elliptic integrals and exhibits similar critical behaviour to the dry wall case

computing the eigenvalues of the generalized sturm liouville problems based on the lie group sl 2 r for the generalized sturm liouville problems we can construct an sl 2 r lie group shooting method to find eigenvalues by using the closure property of the lie group a one step lie group transformation between the boundary values at two ends of the considered interval is established hence we can theoretically derive an analytical characteristic equation to determine the eigenvalues for the generalized sturm liouville problems because the closed form formulas are derived to calculate the unknown left boundary values in terms of l the present method provides an easy numerical implementation and has a cheap computational cost numerical examples are examined to show that the present sl 2 r lie group shooting method is effective

a computational method for solving third order singularly perturbed boundary value problems in this paper a new computational method is presented for solving a class of third order singularly perturbed boundary value problems with a boundary layer at the left of the underlying interval in this method first the given third order singularly perturbed boundary value problem is transformed into a system of two odes subject to suitable initial and boundary conditions and a zeroth order asymptotic expansion for the solution of the given problem is constructed then the reduced terminal value problem is solved analytically in the reproducing kernel space this method is effective and easy to implement a numerical examples is studied to demonstrate the accuracy of the present method results obtained by the method are found to be in good agreement with the exact solution not only in the boundary layer but also away from the layer

rigorous derivation of the perimeter generating function for the mean squared radius of gyration of convex polygons we have rigorously derived the perimeter generating function for the mean squared radius of gyration of convex polygons this function was first conjectured by jensen his nonrigorous result is based on the analysis of the long series expansions

area weighted moments of convex polygons on the square lattice when constructing series expansions for the zero field partition function of the q state potts model on the square lattice using the finite lattice method the series can be extended by one term by adding a correction of q 1 times the number of convex polygons of an appropriate order by virtue of duality such corrections apply for both high and low temperature series for low temperature series corrections to the field dependent partition function are of the same order in the temperature variable as the zero field correction the general corrections are given by area weighted moments of the number of convex polygons the authors obtain conjectured closed form expressions for the two generating functions that give corrections to the first two field derivatives of the low temperature partition function

using reproducing kernel for solving a class of singularly perturbed problems this paper is concerned with a new algorithm for giving the analytical and approximate solutions of a class of partial differential equations with a singularly perturbed term in a reproducing kernel space two numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method indicate the method is simple and effective

on the exhaustive generation of k convex polyominoes the degree of convexity of a convex polyomino p is the smallest integer k such that any two cells of p can be joined by a monotone path inside p with at most k changes of direction in this paper we present a simple algorithm for computing the degree of convexity of a convex polyomino and we show how it can be used to design an algorithm that generates given an integer k all k convex polyominoes of area n in constant amortized time using space o n furthermore by applying few changes we are able to generate all convex polyominoes whose degree of convexity is exactly k

directed compact percolation near a wall iii exact results for the mean length and number of contacts existing exact results for the percolation probability and mean cluster size for compact percolation near a dry wall are extended to the mean cluster length and the mean number of contacts with the wall the results are derived from our previous work on vesicles near an attractive wall and involve elliptic integrals as opposed to the simple rational forms found for the percolation probability and cluster size below the results for the cluster length satisfy previously conjectured differential equations a closed expression is conjectured for the mean size above in terms of a hypergeometric function

exact solution of the convex polygon perimeter and area generating function an explicit expression is derived for the three variable generating function p x y z sigma m or 1 sigma n or 1 sigma r or 1 x2ny2mzrcn m r where cn m r is the number of convex polygons with horizontal width n vertical height m and area r

strip tiling and regular grammars we study the problem of tiling a rectangular p n strip fixed with pieces i e sets of simply connected cells some well known examples are strip tilings with dimers dominoes and or monomers we prove in a constructive way that every tiling problem is equivalent to a regular grammar that is the set of possible tilings constitutes a regular language we propose a straightforward algorithm to transform the tiling problem into its corresponding grammar by means of some standard methods we are then able to obtain some counting generating functions that are rational we go on to give some examples of our method and indicate some of its applications to a number of problems treated in current literature

self avoiding polygons on the square lattice we have developed an improved algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 90 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 529 27 1 biased and the critical exponent 0 500 0005 10 unbiased the critical point is indistinguishable from a root of the polynomial 581x4 7x2 13 0 an asymptotic expansion for the coefficients is given for all n there is strong evidence for the absence of any non analytic correction to scaling exponent

an algorithm for solving nonlinear singular perturbation problems with mechanization in this paper by using the theories and methods of mathematical analysis and computer algebra a reliable algorithm of pade approximation method for solving nonlinear singular perturbation problems was established a new maple procedure spp was established too the procedure spp give not only the approximate analytic solutions but also the numerical solutions of spp six examples are presented to illustrate the implementation of the program the method is very simple and effective for most of singular perturbation problems this would be useful for solving differential equations

generating functions for classes of self avoiding polygons on the square lattice abstract a transfer matrix technique is presented which allows the explicit calculation of the generating functions for those self avoiding polygons on the square lattice which have a perimeter equal to or two larger than the perimeter of their bounding rectangles

enumeration of polyominoes using macsyma this paper shows the use of a symbolic language macsyma to obtain new exact or asymptotic results in combinatorics the examples are taken among polyominoes sic objects the main purpose of the paper is to show how easy it is to bring some methods into operation in order to obtain new results in enumerative combinatorics from the abstract

unidirectional convex polygons on the honeycomb lattice the authors obtain the exact generating function for a class of self avoiding polygons on the honeycomb lattice which are convex in one of the three lattice directions it is shown that the critical exponent alpha of these unidirectional convex polygons is 3 2

sequences of rational torsions on abelian varieties we address the question of how fast the available rational torsion on abelian varieties over increases with dimension the emphasis will be on the derivation of sequences of torsion divisors on hyperelliptic curves work of hellegouarch and lozach and klein may be made explicit to provide sequences of curves with rational torsion divisors of orders increasing linearly with respect to genus the main results in 2 are applications of a new technique which provide sequences of hyperelliptic curves for all torsions in an interval a g a g b g wherea g is quadratic ing andb g is linear ing as well as providing an improvement from linear to quadratic these results provide a wide selection of torsion orders for potential use by those involved in computer integration we conclude by considering possible techniques for divisors of non hyperelliptic curves and for general abelian varieties

convex polyominoes and algebraic languages uses the so called dsv methodology that links some enumeration problems to the theory of algebraic languages to get a system of q difference equations involving the generating functions of convex polyominoes of convex and directed polyominoes and of parallelogram polyominoes according to their height width and area then the author shows various applications of this system

synergistic development of differential approximants and the finite lattice method in lattice statistics several decades of parallel developments in the calculation and analysis of series expansions for lattice statistics have led to many new insights into critical phenomena these studies have centered on the use of the finite lattice method for series expansions in lattice statistics and the use of differential approximants in analysing such series one of these strands of research ultimately led to the result that a number of unsolved lattice statistics problems cannot be expressed as d finite functions somewhat ironically given power and success of differential approximants in analysing series neither the assumed functional form nor any finite generalisation thereof can fit such cases exactly

enumeration of 4 stack polyominoes in this paper we consider the class of 4 stack polyominoes i e polyominoes which can be decomposed into a central rectangle supporting four stack polyominoes one on each side of the rectangle this class of objects recently introduced by marc noy extends the class of centered convex polyominoes and is included into the class of z convex polyominoes using an inclusion exclusion approach we obtain the enumeration of 4 stack polyominoes according to the number of rows and columns moreover we solve some problems posed by marc noy proving that the generating function of 4 stack polyominoes according to the semi perimeter is algebraic and that their asymptotic behavior is n4 n which is immediately smaller than the asymptotic behavior of z convex polyominoes which is n4 n as a corollary of our result we find the generating function of bi centered polyominoes i e convex polyominoes which are both horizontally and vertically centered

on an involution on dyck paths les derniers resultats concernant l enumeration des polyamines parallelogrammes 4 5 8 9 11 14 selon l aire et le nombre de colonnes font apparaitre des fonctions generatrices s exprimant comme rapport de q analogues de fonctions de bessel dans 13 deux interpretations combinatoires de ces memes fonctions sont donnees d une part en termes de multiparcours d arbres et d autre part en termes de multichaines de chemins de dyck a l aide d une involution sur l ensemble d des chemins de dyck d une bijection entre chemins et multiparcours d arbres et d une bijection entre chemins et multichaines nous donnons dans ce papier une preuve bijective de l equivalence de ces deux interpretations

walks on the slit plane in the first part of this paper we enumerate exactly walks on the square lattice that start from the origin but otherwise avoid the half line we call them walks on the slit plane we count them by their length and by the coordinates of their endpoint the corresponding three variable is algebraic of degree 8 moreover for any point i j the length for walks of this type ending at i j is also algebraic of degree 2 or 4 and involves the famous catalan numbers

a parallel algorithm for the enumeration of self avoiding polygons on the square lattice we have developed a parallel algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 110 we have also extended the series for the first 10 area weighted moments and the radius of gyration to 100 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 530 31 3 biased and the critical exponent 0 500 000 1 2 unbiased in addition we obtain very accurate estimates for the leading amplitudes confirming to a high degree of accuracy various predictions for universal amplitude combinations

on an exactly solvable compact cluster growth model on a square lattice we consider a cluster growth model on a square lattice that is in the compact directed percolation universality class the model is exactly solvable as regards its static cluster properties and expressions are given for various quantities of interest such as the mean perimeter length and the mean cluster size the results also provide new information on area perimeter generating functions for a class of self avoiding polygons on square lattices

directed compact percolation near a wall i biased growth the directed compact percolation cluster model of domany and kinzel 1984 is considered in the presence of a wall which is parallel to the growth direction and hence restricts the lateral growth of the cluster in one direction the critical exponents are found to depend on whether the wall is wet or dry in the former case the model is solved exactly for all the standard percolation functions and the critical behaviour is found to be the same as that for cluster growth with no wall present with this boundary condition the cluster is completely attached to the wall and the model may also be viewed as one of symmetric compact cluster growth in the case of a dry wall the cluster may repeatedly leave and return to the wail as it grows and in this case the percolation probability has been derived exactly by lin 1992 and found to have a critical exponent different from that of the bulk lin s result is rederived and an exact formula for the percolation probability is found for a more general model in which the cluster growth is biased either towards or away from the wall it is found that the unbiased case is special in that any bias away from the wall recovers the bulk critical exponent and a bias towards the wall produces a problem in the same class as the wet wall model

perimeter generating functions for the mean squared radius of gyration of convex polygons we have derived long series expansions for the perimeter generating functions of the radius of gyration of various polygons with a convexity constraint using the series we numerically find simple algebraic exact solutions for the generating functions in all cases the size exponent 1

context free grammars differential operators and formal power series abstract in this paper we introduce the concepts of a formal function over an alphabet and a formal derivative based on a set of substitution rules we call such a set of rules a context free grammar because these rules act like a context free grammar in the sense of a formal language given a context free grammar we can associate each formal function with an exponential formal power series in this way we obtain grammatical interpretations of addition multiplication and functional composition of formal power series a surprising fact about the grammatical calculus is that the composition of two formal power series enjoys a very simple grammatical representation we apply this method to obtain simple demonstrations of faa di bruno s formula and some identities concerning bell polynomials stirling numbers and symmetric functions in particular the lagrange inversion formula has a simple grammatical representation from this point of view one sees that cayley s formula on labeled trees is equivalent to the lagrange inversion formula

numerical methods for singular perturbation problems consider the two point boundary value problem for a stiff system of ordinary differential equations an adaptive method to solve these problems even when turning points are present is discussed

explicit determination of nontrivial torsion structures of elliptic curves over quadratic number fields we determine equations of the modular curves x1 n for n 11 13 14 15 16 17 and 18 except for n 17 these are the only existing elliptic or hyperelliptic x1 n applying these x1 n we calculate tables of elliptic curves e over quadratic fields k with torsion groups of one of the following isomorphism types etor k z mz m 11 13 14 15 16 and 18

numerical solutions of linear and nonlinear singular perturbation problems a new method is developed by detecting the boundary layer of the solution of a singular perturbation problem on the non boundary layer domain the singular perturbation problem is dominated by the reduced equation which is solved with standard techniques for initial value problems while on the boundary layer domain it is controlled by the singular perturbation its numerical solution is provided with finite difference methods of which up to sixth order methods are developed the numerical error is maintained at the same level with a constant number of mesh points for a family of singular perturbation problems numerical experiments support the analytical results

two generalizations of column convex polygons column convex polygons were first counted by area several decades ago and the result was found to be a simple rational generating function in this work we generalize that result let a p column polyomino be a polyomino whose columns can have 1 2 p connected components then column convex polygons are equivalent to 1 convex polyominoes the area generating function of even the simplest generalization namely 2 column polyominoes is unlikely to be solvable we therefore define two classes of polyominoes which interpolate between column convex polygons and 2 column polyominoes we derive the area generating functions of those two classes using extensions of existing algorithms the growth constants of both classes are greater than the growth constant of column convex polyominoes rather tight lower bounds on the growth constants complement a comprehensive asymptotic analysis

object grammars and bijections a new systematic approach for the specification of bijections between sets of combinatorial objects is presented it is based on the notion of object grammars object grammars give recursive descriptions of objects and generalize context free grammars the study of a particular substitution in these object grammars confirms once more the key role of dyck words in the domain of enumerative and bijective combinatorics

exact solution of the row convex polygon perimeter generating function an explicit expression is obtained for the perimeter generating function g y sigma n or 2 any2n for row convex polygons on the square lattice where an is the number of 2n step row convex polygons an asymptotic expression for an approximately a mu nn 3 2 is obtained where mu 3 22 and a are given the authors also show that the generating function is an algebraic function and that it satisfies an inhomogeneous linear differential equation of degree three

on directed compact percolation near a damp wall a percolation probability for directed compact percolation near a damp wall which interpolates between the previously examined cases is derived exactly we find that the critical exponent 2 in common with the dry wall rather than the value previously found in the wet wall and bulk cases the solution is found via a mapping to a particular model of directed walks we evaluate the exact generating function for this walk model which is also related to the asep model of traffic flow we compare the underlying mathematical structure of the various cases previously considered and this one by reviewing the common framework of solution via the mapping to different directed walk models

non standard difference schemes for singular perturbation problems revisited non standard finite difference techniques for solving various linear two point singularly perturbed boundary value problems are considered they consist of replacing the original differential equation by a related differential equation with a small deviating argument this related problem is then discretized using finite differences this discretized problem is solved using standard matrix methods the accuracy and robustness of these numerical schemes are then carefully investigated and compared to standard centered finite differences in particular earlier claims for the superiority of such schemes over standard finite differences are shown to be invalid

compact directed percolation with modified boundary rules a forest fire model a modified version of compact directed percolation on a square lattice is examined in the context of a model problem for the spread of forest fires the modification relates to conditioning the extent of the fire spread along the forest boundaries exact expressions are given for the mean perimeter length and the mean size of the damaged forest under such conditioning

directed compact percolation near a wall ii cluster length and size for paper i see ibid vol 27 p 3743 50 1994 the mean cluster size and length for the unbiassed growth of compact clusters near a dry wall are considered in the case of the cluster size below pc an exact expression is obtained for a seed of arbitrary width and distance from the surface it is found that the critical exponent gamma 1 for any finite distance from the surface crossover to the bulk value gamma 2 as the distance from the surface tends to infinity is observed this extends an existing result for the exponent beta of the percolation probability which changes from a value of 2 in the presence of a surface to 1 in the bulk limit the value delta 3 of the scaling size exponent is unchanged by the introduction of the surface the cluster size above pc and the mean cluster length are investigated using differential approximants from which we conjecture that these functions satisfy second order differential equations accepting this conjecture gives a mean size exponent the same as below pc and a logarithmic divergence of the mean length from both sides of the critical point the latter result together with scaling theory predicts that the exponent v sub has the value 2 the same as for the bulk problem

algebraic languages and polyominoes enumeration abstract in this paper the use of algebraic languages theory in solving an open problem in combinatorics is shown by constructing a bijection between convex polyominoes and words of an algebraic language and by solving the corresponding algebraic system we prove that the number of convex polyominoes with perimeter 2n 8 is 2n 11 4 n 4 2n 1 2n n

an operator compact implicit method of exponential type a fourth order accurate tridiagonal operator compact implicit finite difference method for diffusion convection equations is developed the coefficients of the difference operator contain exponentials of the coefficients of the differential operator in this way the method avoids spurious oscillations when the cell reynolds number is large numerical results and comparisons to other methods are presented

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

mechanization for solving spp by reducing order method in this paper by using the theories and methods of mathematical analysis and computer algebra a reliable algorithm of reduction of order approximation method for solving singular perturbation problems was established a new maple procedure redordproc was established too the procedure redordproc give not only the approximate analytic solutions but also the numerical solutions of the kinetics problems some examples are presented to illustrate the implementation of the program

statistical mechanics of directed models of polymers in the square lattice directed square lattice models of polymers and vesicles have received considerable attention in the recent mathematical and physical sciences literature these are idealized geometric directed lattice models introduced to study phase behaviour in polymers and include dyck paths partially directed paths directed trees and directed vesicles models directed models are closely related to models studied in the combinatorics literature and are often exactly solvable they are also simplified versions of a number of statistical mechanics models including the self avoiding walk lattice animals and lattice vesicles the exchange of approaches and ideas between statistical mechanics and combinatorics have considerably advanced the description and understanding of directed lattice models and this will be explored in this review the combinatorial nature of directed lattice path models makes a study using generating function approaches most natural in contrast the statistical mechanics approach would introduce partition functions and free energies and then investigate these using the general framework of critical phenomena generating function and statistical mechanics approaches are closely related for example questions regarding the limiting free energy may be approached by considering the radius of convergence of a generating function and the scaling properties of thermodynamic quantities are related to the asymptotic properties of the generating function in this review the methods for obtaining generating functions and determining free energies in directed lattice path models of linear polymers is presented these methods include decomposition methods leading to functional recursions as well as the temperley method that is implemented by creating a combinatorial object one slice at a time a constant term formulation of the generating function will also be reviewed the thermodynamic features and critical behaviour in models of directed paths may be informative about the underlying properties that determine phase diagrams for wider classes of models including physical models of polymers of particular interest are adsorption and collapse transitions in models of polymers and copolymers the properties of thermodynamic quantities in those models are described by tricritical scaling this is reviewed for directed path models and the generating function approaches can be used to apply tricritical scaling to models of adsorbing inflating and collapsing directed lattice paths critical exponents for a variety of models can be obtained in this manner and with it a better understanding and a classification of the models

stable high order methods for differential equations with small coefficients for the second order terms abstract stable high order methods are developed for solving differential equations with small coefficients for the second order terms in particular second fourth and sixth order methods are developed the methods are developed for general linear two point boundary value problems and proved to be stable and accurate results are presented for the one dimensional convection diffusion equation involving problems with and without boundary layers the methods converged for all values of parameters attempted and the results compared favorably with other methods the methods developed in this paper are accurate stable easy to use and applicable to other problems

statistics of collapsed lattice animals rigorous results and monte carlo simulations in this paper a two dimensional model of collapsed lattice animals with nearest neighbour contact interactions is introduced and the limiting free energy for the model is proved to exist we also establish that the partition function for collapsed lattice animals is related to the reliability polynomial for the m m grid graph and to the generating function for spanning forests on the m m grid graph from these relationships and using results about directed animals we are able to obtain rigorous upper and lower bounds on the free energy for collapsed lattice animals in addition we present monte carlo evidence that the heat capacity associated with this model remains bounded in the thermodynamic limit in fact we do not find any evidence for a second order phase transition for collapsed lattice animals

rigorous result for the area weighted moments of convex polygons on the square lattice a generating function for the area weighted moments of the number of convex polygons on the square lattice is derived rigorously a special case of the authors result was obtained first by enting and guttmann 1988 nonrigorously their calculation is based on series expansions

a strip like tiling algorithm we extend our previous results on the connection between strip tiling problems and regular grammars by showing that an analogous algorithm is applicable to other tiling problems not necessarily related to rectangular strips we find generating functions for monomer and dimer tilings of t and l shaped figures holed and slotted strips diagonal strips and combinations of them and show how analogous results can be obtained by using different pieces

the perimeter generating functions of three choice imperfect and one punctured staircase polygons we consider the isotropic perimeter generating functions of three choice imperfect and one punctured staircase polygons whose 8th order linear fuchsian odes are previously known we derive simple relationships between the three generating functions and show that all three generating functions are joint solutions of a common 12th order fuchsian linear ode we find that the 8th order differential operators can each be rewritten as a direct sum of a direct product with operators no larger than 3rd order we give closed form expressions for all the solutions of these operators in terms of 2 f 1 hypergeometric functions with rational and algebraic arguments the solutions of these linear differential operators can in fact be expressed in terms of two modular forms since these 2 f 1 hypergeometric functions can be expressed with two rational or algebraic pullbacks