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

polygons on the honeycomb lattice the authors enumerated the number of self avoiding polygons on the honeycomb lattice to 82 steps analysis of the resulting series gives for the connective constant mu 1 8477594 or 6 10 6 in agreement with nienhuis value 1984 of mu 2 2 12 1 84775906 the critical exponent alpha 0 4999 or 10 4 unbiased or alpha 0 49998 or 2 10 0 5 biased a non physical singularity on the negative real axis is accurately located and some curious results concerning the confluent singularity are obtained

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

automatic average case analysis of algorithms abstract many probabilistic properties of elementary discrete combinatorial structures of interest for the average case analysis of algorithms prove to be decidable this paper presents a general framework in which such decision procedures can be developed it is based on a combination of generating function techniques for counting and complex analysis techniques for asymptotic estimations we expose here the theory of exact analysis in terms of generating functions for four different domains the iterative recursive and unlabelled labelled data type domains we then present some major components of the associated asymptotic theory and exhibit a class of naturally arising functions that can be automatically analyzed a fair fragment of this theory is also incorporated into a system called lambda upsilon omega in this way using computer algebra one can produce automatically non trivial average case analyses of algorithms operating over a variety of decomposable combinatorial structures at a fundamental level this paper is part of a global attempt at understanding why so many elementary combinatorial problems tend to have elementary asymptotic solutions in several cases it proves possible to relate entire classes of elementary combinatorial problems whose structure is well defined with classes of elementary special functions and classes of asymptotic forms relative to counting probabilities or average case complexity

reconstruction of convex lattice sets from tomographic projections in quartic time filling operations are procedures which are used in discrete tomography for the reconstruction of lattice sets having some convexity constraints many algorithms have been published giving fast implementations of these operations and the best running time s brunetti a daurat a kuba fast filling operations used in the reconstruction of convex lattice sets in proc of dgci 2006 in lecture notes in comp sci vol 4245 2006 pp 98 109 is o n 2logn time where n is the size of projections in this paper we improve this result by providing an implementation of the filling operations in o n 2 as a consequence we reduce the time complexity of the reconstruction algorithms for many classes of lattice sets having some convexity properties in particular the reconstruction of convex lattice sets satisfying the conditions of gardner gritzmann r j gardner p gritzmann discrete tomography determination of finite sets by x rays trans amer math soc 349 1997 2271 2295 can be performed in o n 4 time

directed compact percolation cluster size and hyperscaling exact recurrence relations are obtained for the length and size distributions of compact directed percolation clusters on the square lattice the corresponding relation for the moment generating function of the length distribution is obtained in closed form whereas in the case of the size distribution only the first three moments are obtained the work is carried out for clusters grown from a seed of arbitrary width on an anisotropic lattice a duality property is shown to exist which relates the moment generating functions on the two sides of the critical curve the moments of both distributions have critical exponents which satisfy a constant gap hypothesis with gap exponents nu 2 and delta 3 corresponding to the scaling length and scaling size respectively the usual hyperscaling relation for directed percolation is found to be invalid for compact clusters and is replaced by delta nu d nu perpendicular to where nu perpendicular to is the exponent corresponding to the cluster width and d is the number of transverse dimensions 1 for the square lattice

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

a framework for generating some discrete sets with disjoint components by using uniform distributions discrete tomography deals with the reconstruction of discrete sets from few projections assuming that the set to be reconstructed belongs to a certain class of discrete sets with some geometrical properties is a commonly used technique to reduce the number of possibly many different solutions of the same reconstruction problem the average performance of reconstruction algorithms are often tested on such classes by choosing elements of a given class from uniform random distributions this paper presents a general framework for generating discrete sets with disjoint connected components using uniform distributions especially the uniform random generation of hv convex discrete sets and q convex discrete sets according to the size of the minimal bounding rectangle are discussed

the existence of refinement mappings abstract refinement mappings are used to prove that a lower level specification correctly implements a higher level one we consider specifications consisting of a state machine which may be infinite state that specifies safety requirements and an arbitrary supplementary property that specifies liveness requirements a refinement mapping from a lower level specification s 1 to a higher level one s 2 is a mapping from s 1 s state space to s 2 s state space it maps steps of s 1 s state machine to steps of s 2 s state machine and maps behaviors allowed by s 1 to behaviors allowed by s 2 we show that under reasonable assumptions about the specification if s 1 implements s 2 then by adding auxiliary variables to s 1 we can guarantee the existence of a refinement mapping this provides a completeness result for a practical hierarchical specification method

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

geometric properties of matrices induced by pattern avoidance the notion of submatrix avoidance in polyominoes has recently been introduced in 2 with the aim of extending most of the concepts and properties concerning pattern avoiding permutations to the setting of polyominoes in this paper we use submatrix avoidance to describe families of polyominoes which in the literature are usually defined by means of the geometric constraints of convexity k convexity and directedness to reach this goal we provide an extension of the notion of pattern in a polyomino by introducing generalized polyomino patterns in the second part of the paper we tackle the same problem in the context of discrete sets which can be naturally regarded as binary matrices in this case we consider two types of geometric constraints convexity and directedness and we study how these constraints can be imposed on matrices by using submatrix avoidance

comparison of algorithms for reconstructing hv convex discrete sets three reconstruction algorithms to be used for reconstructing hv convex discrete sets from their row and column sums are compared all these algorithms have two versions one for reconstructing hv convex polyominoes and another one for reconstructing hv convex 8 connected discrete sets in both classes of discrete sets the algorithms are compared from the viewpoints of average execution time and memory complexities discrete sets with given sizes are generated with uniform distribution and then reconstructed from their row and column sums first we have implemented two previously published algorithms according to our comparisons the algorithm which was better from the viewpoint of worst time complexity was the worse from the viewpoint of average time complexity and memory requirements then as a new method a combination of the two algorithms was also implemented and it is shown that it inherits the best properties of the other two methods

reconstruction of 4 and 8 connected convex discrete sets from row and column projections in this paper we examine the problem of reconstructing a discrete two dimensional set from its two orthogonal projection h v when the set satisfies some convexity conditions we show that the algorithm of the paper int j imaging systems and technol 9 1998 69 is a good heuristic algorithm but it does not solve the problem for all h v instances we propose a modification of this algorithm solving the problem for all h v instances by starting to build the spine the complexity of our reconstruction algorithm is o mn log mn min m2 n2 in the worst case however according to our experimental results in 99 of the studied cases the algorithm is able to reconstruct a solution without using the newly introduced operation in such cases the upper bound of the complexity of the algorithm is o mn log mn a systematic comparison of this algorithm was done and the results show that this algorithm has the better average complexity than other published algorithms the way of comparison and the results are given in a separate paper linear algebra appl submitted finally we prove that the problem can be solved in polynomial time also in a class of discrete sets which is larger than the class of convex polyominoes namely in the class of 8 connected convex sets

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

grunbaum s inequality for projections let k be a convex body in rnrn whose centroid is at the origin let e g n k e g n k be a subspace and let sn 1 sn 1 we find the best constant c kn 1 k so that k e k c k e k and completely determine the minimizer here k k is k dimensional volume k ek e is the projection of k onto e and x rn x 0 our result generalizes both grunbaum s inequality and an old inequality of minkowski and radon

reconstructing convex polyominoes from horizontal and vertical projections in 1 we studied the problem of reconstructing a discrete set 5 from its horizontal and vertical projections we defined an algorithm that establishes the existence of a convex polyomino whose horizontal and vertical projections are equal to a pair of assigned vectors h v with h m and v n its computational cost is o n4m4 in this paper we introduce some operations for recontructing convex polyominoes by means of vectors h s and v s partial sums these operations allows us to define a new algorithm whose complexity is less than o n2m2

enumeration of l convex polyominoes by rows and columns in this paper we consider the class of l convex polyominoes i e the convex polyominoes in which any two cells can be connected by a path of cells in the polyomino that switches direction between the vertical and the horizontal at most once using the eco method we prove that the number fn of l convex polyominoes with perimeter 2 n 2 satisfies the rational recurrence relation fn 4fn 1 2fn 2 with f0 1 f1 2 f2 7 moreover we give a combinatorial interpretation of this statement in the last section we present some open problems

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

solving nonograms by combining relaxations nonograms also known as japanese puzzles are a specific type of logic drawing puzzles the challenge is to fill a grid with black and white pixels in such a way that a given description for each row and column indicating the lengths of consecutive segments of black pixels is adhered to although the nonograms in puzzle books can usually be solved by hand the general problem of solving nonograms is np hard in this paper we propose a reasoning framework that can be used to determine the value of certain pixels in the puzzle given a partial filling constraints obtained from relaxations of the nonogram problem are combined into a 2 satisfiability 2 sat problem which is used to deduce pixel values in the nonogram solution by iterating this procedure starting from an empty grid it is often possible to solve the puzzle completely all the computations involved in the solution process can be performed in polynomial time our experimental results demonstrate that the approach is capable of solving a variety of nonograms that cannot be solved by simple logic reasoning within individual rows and columns without resorting to branching operations in addition we present statistical results on the solvability of nonograms obtained by applying our method to a large number of nonograms

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

enumeration of self avoiding walks on the square lattice we describe a new algorithm for the enumeration of self avoiding walks on the square lattice using up to 128 processors on a hp alpha server cluster we have enumerated the number of self avoiding walks on the square lattice to length 71 series for the metric properties of mean square end to end distance mean square radius of gyration and mean square distance of monomers from the end points have been derived to length 59 an analysis of the resulting series yields accurate estimates of the critical exponents and confirming predictions of their exact values likewise we obtain accurate amplitude estimates yielding precise values for certain universal amplitude combinations finally we report on an analysis giving compelling evidence that the leading non analytic correction to scaling exponent 1 3 2

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

a temporal logic for proving properties of topologically general executions we present a generalization of the temporal propositional logic of linear time which is useful for stating and proving properties of the generic execution sequence of a parallel program or a non deterministic program the formal system we present is exactly that same as the third of three logics presented by lehmann and shelah information and control53 165 198 1982 but we give it a different semantics the models are tree models of arbitrary size similar to those used in branching time temporal logic the formulation we use allows us to state properties of the co meagre family of paths where the term co meagre refers to a set whose complement is of the first category in baire s classification looking at the set of paths in the model as a metric space our system is decidable sound and complete for models of arbitrary size but it has the finite model property namely every sentence having a model has a finite model

stacking models of vesicles and compact clusters we investigate three simple lattice models of two dimensional vesicles these models differ in their behavior from the universality class of partially convex polygons which has been recently established they do not have the tricritical scaling of those models and furthermore display a surprising feature their perimeter free energy is discontinuous with an isolated value at zero pressure we give the full asymptotic descriptions of the generating functions in area and perimeter variables from theq series solutions and obtain the scaling functions where applicable

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

inflating square and rectangular lattice vesicles in this paper two simple models of lattice vesicles in two and in three dimensions are examined lattice vesicles have a rich mathematical nature and the combinatorial and scaling properties of square and rectangular lattice vesicle models are investigated and then generalized to three dimensions the phase diagram of the vesicles also includes a multicritical point and scaling of the generating functions of these models are studied to test the models for tricritical scaling behaviour in the models of square vesicles in the square lattice and cubical vesicles in the cubic lattice the tricritical scaling forms are explicitly confirmed however rectangular vesicles in the square lattice and rectangular box vesicles in the cubic lattice appear not to conform to the expected tricritical scaling forms

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

the complexity of probabilistic verification we determine the complexity of testing whether a finite state sequential or concurrent probabilistic program satisfies its specification expressed in linear time temporal logic for sequential programs we present an algorithm that runs in time linear in the program and exponential in the specification and also show that the problem is in pspace matching the known lower bound for concurrent programs we show that the problem can be solved in time polynomial in the program and doubly exponential in the specification and prove that it is complete for double exponential time we also address these questions for specifications described by automata or formulas in extended temporal logic

random generation of q convex sets the problem of randomly generating q convex sets is considered we present two generators the first one uses the q convex hull of a set of random points in order to generate a q convex set included in the square 0 n 2 this generator is very simple but is not uniform and its performance is quadratic in n the second one exploits a coding of the salient points which generalizes the coding of a border of polyominoes it is uniform and is based on the method by rejection experimentally this algorithm works in linear time in the length of the word coding the salient points besides concerning the enumeration problem we determine an asymptotic formula for the number of q convex sets according to the size of the word coding the salient points in a special case and in general only an experimental estimation

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

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

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

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

critical exponents from nonlinear functional equations for partially directed cluster models we present a method for the derivation of the generating function and computation of critical exponents for several cluster models staircase bar graph and directed column convex polygons as well as partially directed self avoiding walks starting with nonlinear functional equations for the generating function by linearizing these equations we first give a derivation of the generating functions the nonlinear equations are further used to compute the thermodynamic critical exponents via a formal perturbation ansatz alternatively taking the continuum limit leads to nonlinear differential equations from which one can extract the scaling function we find that all the above models are in the same universality class with exponents y 1 2 7 1 3 and b 2 3 all models have as their scaling function the logarithmic derivative of the airy function

directed paths in a wedge models of directed paths have been used extensively in the scientific literature to model linear polymers in this paper we examine a directed path model of a linear polymer in a confining geometry a wedge we are particularly interested in cn the number of directed lattice paths of length n steps which take steps in the north east and south east directions and are confined in the wedge y x p where p is an integer we examine the case p 2 in detail and show that the generating function satisfies a functional equation with quartic kernel we give a formal solution by using the kernel method show that this model is not d finite and determine asymptotics to leading order for cn in particular the number of paths of length n in the wedge y x 2 is given by where the constant 0 67874 can be determined to arbitrary accuracy with little computational effort

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 fairness and randomness we investigate the relation between the behavior of non deterministic systems under fairness constraints and the behavior of probabilistic systems to this end first a framework based on computable stopping strategies is developed that provides a common foundation for describing both fair and probabilistic behavior on the basis of stopping strategies it is then shown that fair behavior corresponds in a precise sense to random behavior in the sense of martin lof s definition of randomness we view probabilistic systems as concrete implementations of more abstract non deterministic systems under this perspective the question is investigated what probabilistic properties are needed in such an implementation to guarantee with probability one certain required fairness properties in the behavior of the probabilistic system generalizing earlier concepts of e bounded transition probabilities we introduce the notion of divergent probabilistic systems which enables an exact characterization of the fairness properties of a probabilistic implementation looking beyond pure fairness properties we also investigate what other qualitative system properties are guaranteed by probabilistic implementations of fair non deterministic behavior this leads to a completeness result which generalizes a well known theorem by pnueli and zuck

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

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

grunbaum s inequality for sections abstract we show e f x d x e f x d x k 1 n 1 1 k 1 for all k dimensional subspaces e r n e s n 1 and all concave functions f r n 0 with 0 0 r n f x d x and r n x f x d x at the origin o r n here x x 0 as a consequence of this result we get the following generalization of grunbaum s inequality vol k k e vol k k e k n 1 k for all convex bodies k r n with centroid at the origin k dimensional subspaces e r n and e s n 1 the lower bounds in both of our inequalities are the best possible and we discuss the equality conditions

a transfer matrix approach to the enumeration of plane meanders a closed plane meander of order n is a closed self avoiding curve intersecting an infinite line 2n times meanders are considered distinct up to any smooth deformation leaving the line fixed we have developed an improved algorithm based on transfer matrix methods for the enumeration of plane meanders while the algorithm has exponential complexity its rate of growth is much smaller than that of previous algorithms the algorithm is easily modified to enumerate various systems of closed meanders semi meanders open meanders and many other geometries

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

anisotropic step mutual contact and area weighted festoons and parallelogram polyominoes on the triangular lattice we present results for the generating functions of polygons and more general objects that can touch constructed from two fully directed walks on the infinite triangular lattice enumerated according to each type of step and weighted proportional to the area and the number of contacts between the directed sides of the objects in general these directed objects are known as festoons being constructed from the so called friendly directed walks while the subset constructed from vicious walks are staircase polygons also known as parallelogram polyominoes additionally we give explicit formulae for various first area moment generating functions that is when the area is summed over all configurations with a given perimeter these results generalize and summarize nearly all known results on the square lattice since such results can be obtained by setting one of the step weights to zero all our results for the triangular lattice are new and hence provide the opportunity to study subtle changes in scaling between lattices in most cases we give our results both in terms of ratios of infinite q series and as continued fractions

discrete tomography determination of finite sets by x rays we study the determination of finite subsets of the integer lattice en n 2 by x rays in this context an x ray of a set in a direction u gives the number of points in the set on each line parallel to u for practical reasons only x rays in lattice directions that is directions parallel to a nonzero vector in the lattice are permitted by combining methods from algebraic number theory and convexity we prove that there are four prescribed lattice directions such that convex subsets of en i e finite subsets f with f en n conv f are determined among all such sets by their x rays in these directions we also show that three x rays do not suffice for this purpose this answers a question of larry shepp and yields a stability result related to hammer s x ray problem we further show that any set of seven prescribed mutually nonparallel lattice directions in 22 have the property that convex subsets of 22 are determined among all such sets by their x rays in these directions we also consider the use of orthogonal projections in the interactive technique of successive determination in which the information from previous projections can be used in deciding the direction for the next projection we obtain results for finite subsets of the integer lattice and also for arbitrary finite subsets of euclidean space which are the best possible with respect to the numbers of projections used

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

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

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

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

attribute grammars are useful for combinatorics abstract the purpose of this paper is to show the use of attribute grammars in solving some combinatorics problems for example we give the generating function for the shape of skew ferrers diagrams according to the number of cells and the number of columns this result is new and proves that skew ferrer diagrams are related to new basic bessel functions

reasoning with time and chance the temporal propositional logic of linear time is generalized to an uncertain world in which random events may occur the formulas do not mention probabilities explicitly i e the only probability appearing explicitly in formulas is probability one this logic is claimed to be useful for stating and proving properties of probabilistic programs it is convenient for proving those properties that do not depend on the specific distribution of probabilities used in the program s random draws the formulas describe properties of execution sequences the models are stochastic systems with state transition probabilities three different axiomatic systems are proposed and shown complete for general models finite models and models with bounded transition probabilities respectively all three systems are decidable by the results of rabin trans amer math soc 141 1969 1 35

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

determination of q convex sets by x rays in this paper the problem of the determination of lattice sets from x rays is studied we define the class of q convex sets along a set d of directions which generalizes classical lattice convexity and we prove that for any d the x rays along d determine all the convex sets if and only if it determines all the q convex sets along d as a consequence any algorithm which reconstructs q convex sets from x rays can be used to reconstruct convex lattice sets from x rays along directions which provide uniqueness this gives a constructive answer to the discrete version of hammer s x ray problem

watermelon uniform random generation with applications watermelons are particular configurations of vicious walkers in these configurations each path starts and ends at the same ordinate we present a simple uniform random generation algorithm of watermelons based on enumeration formulas of star configurations with or without a wall the performance of this algorithm is better than earlier ones in the case of watermelons with few walkers using appropriate bijections these algorithms can also generate underdiagonal paths realizers or schnyder trees twin parallelogram polyominoes according to their perimeter and width baxter permutations according to the number of rises etc moreover we present some experimental results on the height of watermelons and realizers

perimeter generating function for row convex polygons on the rectangular lattice the perimeter generating function derived recently by brak et al 1990 for row convex polygons on the square lattice is generalized to the rectangular lattice

a calculus for the random generation of labelled combinatorial structures a systematic approach to the random generation of labelled combinatorial objects is presented it applies to structures that are decomposable i e formally specifiable by grammars involving set sequence and cycle constructions a general strategy is developed for solving the random generation problem with two closely related types of methods for structures of size n the boustrophedonic algorithms exhibit a worst case behaviour of the form o n log n the sequential algorithms have worst case o n2 while offering good potential for optimizations in the average case the complexity model is in terms of arithmetic operations and both methods appeal to precomputed numerical table of linear size that can be computed in time o n2 a companion calculus permits systematically to compute the average case cost of the sequential generation algorithm associated to a given specification using optimizations dictated by the cost calculus several random generation algorithms of the sequential type are developed most of them have expected complexity 1 2n log n and are thus only slightly superlinear the approach is exemplified by the random generation of a number of classical combinatorial structures including cayley trees hierarchies the cycle decomposition of permutations binary trees functional graphs surjections and set partitions

defining fairness in reactive and concurrent systems we define when a linear time temporal property is a fairness property with respect to a given system this captures the essence shared by most fairness assumptions that are used in the specification and verification of reactive and concurrent systems such as weak fairness strong fairness k fairness and many others we provide three characterizations of fairness a language theoretic a game theoretic and a topological characterization it turns out that the fairness properties are the sets that are large from a topological point of view that is they are the co meager sets in the natural topology of runs of a given system this insight provides a link to probability theory where a set is large when it has measure 1 while these two notions of largeness are similar they do not coincide in general however we show that they coincide for regular properties and bounded borel measures that is an regular temporal property of a finite state system has measure 1 under a bounded borel measure if and only if it is a fairness property with respect to that system the definition of fairness leads to a generic relaxation of correctness of a system in linear time semantics we define a system to be fairly correct if there exists a fairness assumption under which it satisfies its specification equivalently a system is fairly correct if the set of runs satisfying the specification is topologically large we motivate this notion of correctness and show how it can be verified in a system

osculating and neighbour avoiding polygons on the square lattice we study two simple modifications of self avoiding polygons saps osculating polygons ops are a super set in which we allow the perimeter of the polygon to touch at a vertex neighbour avoiding polygons naps are only allowed to have nearest neighbour vertices provided these are joined by the associated edge and thus form a sub set of saps we use the finite lattice method to count the number of ops and naps on the square lattice we also calculate their radius of gyration and the first area weighted moment analysis of the series confirms exact predictions for the critical exponents and the universality of various amplitude combinations for both cases we have found exact solutions for the number of convex and almost convex polygons

three attractive osculating walkers and a polymer collapse transition consider n interacting lock step walkers in one dimension which start at the points 0 2 4 2 n 1 and at each tick of a clock move unit distance to the left or right with the constraint that if two walkers land on the same site their next steps must be in the opposite direction so that crossing is avoided when two walkers visit and then leave the same site an osculation is said to take place the space time paths of these walkers may be taken to represent the configurations of n fully directed polymer chains of length t embedded on a directed square lattice if a weight is associated with each of the i osculations the partition function is z t n lambda sum nolimits i 0 left lfloor tfrac n 1 t 2 right rfloor z t i n lambda i where z n t i is the number of t step configurations having i osculations when 0 the partition function is asymptotically equal to the number of vicious walker star configurations for which an explicit formula is known the asymptotics of such configurations was discussed by fisher in his boltzmann medal lecture also for n 2 the partition function for arbitrary is easily obtained by fisher s necklace method for n 2 and 0 the only exact result so far is that of guttmann and voge who obtained the generating function g n lambda u equiv sum nolimits t 0 infty z t n lambda u t for 1 and n 3 the main result of this paper is to extend their result to arbitrary by fitting computer generated data it is conjectured that z 3 t satisfies a third order inhomogeneous difference equation with constant coefficients which is used to obtain g n lambda u frac lambda 3 lambda 2 lambda 12 5 lambda lambda 2 u 2 lambda 3 u 2 2 lambda 4 lambda 2 u 2 1 text c 2u lambda 2 lambda 2 u lambda 1 4 lambda u 4 lambda 2 u 2 where c u tfrac 1 sqrt 1 4u 2u the generating function for catalan numbers the nature of the collapse transition which occurs at 4 is discussed and extensions to higher values of n are considered it is argued that the position of the collapse transition is independent of n

exact solution of the staircase and row convex polygon perimeter and area generating function an explicit expression is obtained for the perimeter and area generating function g y z sigma n or 2 sigma m or 1 cn mynzm where cn m is the number of row convex polygons with area m and perimeter n a similar expression is obtained for the area perimeter generating function for staircase polygons both expressions contain q series

an algorithm reconstructing convex lattice sets in this paper we study the problem of reconstructing special lattice sets from x rays in a finite set of prescribed directions we present the class of q convex sets which is a new class of subsets of z2 having a certain kind of weak connectedness the main result of this paper is a polynomial time algorithm solving the reconstruction problem for the q convex sets these sets are uniquely determined by certain finite sets of directions as a result this algorithm can be used for reconstructing convex subsets of z2 from their x rays in some suitable sets of four lattice directions or in any set of seven mutually nonparallel lattice directions

fairness distances and degrees abstract we show the identity between sets of fair computations in recursive transition graphs sets of cluster points of finite computations for 01 ultra metrics refining the baire metrics and 03 subsets of the results are applied to recursive marked trees fairness definitions regular languages and 03 sets

basic analytic combinatorics of directed lattice paths this paper develops a unified enumerative and asymptotic theory of directed two dimensional lattice paths in half planes and quarter planes the lattice paths are specified by a finite set of rules that are both time and space homogeneous and have a privileged direction of increase they are then essentially one dimensional objects the theory relies on a specific kernel method that provides an important decomposition of the algebraic generating functions involved as well as on a generic study of singularities of an associated algebraic curve consequences are precise computable estimates for the number of lattice paths of a given length under various constraints bridges excursions meanders as well as a characterization of the limit laws associated to several basic parameters of paths

on the area of square lattice polygons we consider the generating function of self avoiding square lattice polygons grouped by both area and perimeter the generating function for polygons of arean is found to diverge atx c 0 251834 with an exponent of zero the mean perimeter of polygons with arean is found to be proportional ton while the mean area of polygons with perimetern is found to be proportional ton 1 5

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