applications of riordan matrix functions to bernoulli and euler polynomials abstract we define riordan matrix functions associated with riordan arrays and study their algebraic properties we also give their applications in the construction of new classes of bernoulli and euler polynomials and bernoulli and euler numbers referred to as the duals and conjugate bernoulli and euler polynomials and dual and conjugate bernoulli and euler numbers respectively

deviations from uniformity in random strings we show that almost all binary strings of length n contain all blocks of size 1 e log2n a close to uniform number of times from this we derive tight bounds on the discrepancy of random infinite strings our results are obtained through explicit generating function expressions and contour integration estimates

row polynomial matrices of riordan arrays abstract let r r n k n k 0 be a riordan array define the row polynomials r n q k 0 n r n k q k and the row polynomial matrix r q r n k q n k 0 by r n k q j k n r n j q j k then r q is also a riordan array with the r n q located on the leftmost column of r q in this paper we investigate combinatorial properties of the matrix r q and the sequence r n q n 0 including their characterizations the q total positivity of r q and the q log convexity of r n q n 0

row sums and alternating sums of riordan arrays abstract here we use row sum generating functions and alternating sum generating functions to characterize riordan arrays and subgroups of the riordan group numerous applications and examples are presented which include the construction of girard waring type identities we also show the extensions to weighted sum generating functions called the expected value generating functions of riordan arrays

directed abelian algebras and their application to stochastic models with each directed acyclic graph this includes some d dimensional lattices one can associate some abelian algebras that we call directed abelian algebras daas on each site of the graph one attaches a generator of the algebra these algebras depend on several parameters and are semisimple using any daa one can define a family of hamiltonians which give the continuous time evolution of a stochastic process the calculation of the spectra and ground state wave functions stationary state probability distributions is an easy algebraic exercise if one considers d dimensional lattices and chooses hamiltonians linear in the generators in finite size scaling the hamiltonian spectrum is gapless with a critical dynamic exponent z d one possible application of the daa is to sandpile models in the paper we present this application considering one and two dimensional lattices in the one dimensional case when the daa conserves the number of particles the avalanches belong to the random walker universality class critical exponent ensuremath sigma ensuremath tau 3 2 we study the local density of particles inside large avalanches showing a depletion of particles at the source of the avalanche and an enrichment at its end in two dimensions we did extensive monte carlo simulations and found ensuremath sigma ensuremath tau 1 780 ifmmode pm else textpm fi 0 005

directed fixed energy sandpile model we numerically study the directed version of the fixed energy sandpile on a closed square lattice the dynamical evolution of a fixed density of sand grains is studied the activity of the system shows a continuous phase transition around a critical density while the deterministic version has the set of nontrivial exponents the stochastic model is characterized by mean field like exponents

on the characterizing properties of the permanental polynomials of graphs abstract let g be a graph and a g the adjacency matrix of g the permanental polynomial of g is defined as g x per x i a g if two graphs g and h have the same permanental polynomial then g is called a per cospectral mate of h a graph g is said to be characterized by its permanental polynomial if all the per cospectral mates of g are isomorphic to g it is shown that complete graphs stars regular complete bipartite graphs and odd cycles are characterized by their permanental polynomials we prove that in general the permanental polynomial cannot characterize the paths and even cycles in particular for each l 1 and m 2 we can find non isomorphic per cospectral mates of p 4 l 3 and c 4 m respectively when we restrict our consideration to connected graphs both the paths and even cycles c 4 l 2 are characterized by their permanental polynomials

the elements of finite order in the riordan group over the complex field abstract an element of finite order in the riordan group over the real field must have order 1 or 2 if we extend all the entries to be complex numbers then it may have any finite order in the present paper we investigate the elements of finite order of the riordan group over the complex field this notion leads us to define an element of pseudo order k 2 and k pseudo involution respectively it turns out that the inverse of a k pseudo involution only differs from it in signs we clarify some relationship between the elements of pseudo order k and k pseudo involutions in particular k pseudo involutions for k 0 mod 4 are characterized by a single sequence the subgroups of the riordan group formed by the elements having pseudo order of a prime power p r are also introduced

umbral presentations for polynomial sequences abstract using random variables as motivation this paper presents an exposition of formalisms developed in 1 2 for the classical umbral calculus a new construction and a variety of examples are presented culminating in several descriptions of sequences of binomial type in terms of umbral polynomials

universality classes in directed sandpile models we perform large scale numerical simulations of a directed version of the two state stochastic sandpile model numerical results show that this stochastic model defines a new universality class with respect to the abelian directed sandpile the physical origin of the different critical behaviour has to be ascribed to the presence of multiple topplings in the stochastic model these results provide new insight into the long debated question of universality in abelian and stochastic sandpiles

a characterization of graphs with rank 5 abstract the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph g with rank 5 or equivalently what is the structure of a connected n vertex graph g whose adjacency matrix has nullity n 5 in this paper we completely characterize connected graphs g whose adjacency matrix has rank 5

confirming and extending the hypothesis of universality in sandpiles stochastic sandpiles self organize to an absorbing state critical point with scaling behavior different from directed percolation dp and characterized by the presence of an additional conservation law this is usually called the c dp or manna universality class there remains however an exception to this universality principle a sandpile automaton introduced by maslov and zhang which was claimed to be in the dp class despite the existence of a conservation law we show by means of careful numerical simulations as well as by constructing and analyzing a field theory that contrarily to what was previously thought this sandpile is also in the c dp or manna class this confirms the hypothesis of universality for stochastic sandpiles and gives rise to a fully coherent picture of self organized criticality in systems with conservation in passing we obtain a number of results for the c dp class and introduce a strategy to easily discriminate between dp and c dp scaling

permanental polynomials of graphs abstract the first section surveys recent results on the permanental polynomial of a square matrix a i e per xi a the second section concerns the permanental polynomial of the adjacency matrix of a graph the final section is an introduction to the permanental polynomial of the laplacian matrix of a graph an appendix lists some of these latter polynomials

compact recognizers of episode sequences given two strings x a1 an and p b1 bm over an alphabet the problem of testing whether p occurs as a subsequence of x is trivially solved in linear time it is also known that a simple o n log time preprocessing of x makes it easy to decide subsequently for any p and in at most p log character comparisons whether p is a subsequence of x these problems become more complicated if one asks instead whether p occurs as a subsequence of some substring y of x of bounded length this paper presents an automaton built on the textstring x and capable of identifying all distinct minimal substrings y of x having p as a subsequence by a substring y being minimal with respect to p it is meant that p is not a subsequence of any proper substring of y for every minimal substring y the automaton recognizes the occurrence of p having the lexicographically smallest sequence of symbol positions in y it is not difficult to realize such an automaton in time and space o n2 for a text of n characters one result of this paper consists of bringing those bounds down to linear or o n log n respectively depending on whether the alphabet is bounded or of arbitrary size thereby matching the corresponding complexities of automata constructions for offline exact string searching having built the automaton the search for all lexicographically earliest occurrences of p in x is carried out in time o i 1mrocci i or o n i 1mrocci i log n depending on whether the alphabet is fixed or arbitrary where rocci is the number of distinct minimal substrings of x having b1 bi as a subsequence note that each such substring may occur many times in x but is counted only once in the bound all log factors appearing in the above bounds can be further reduced to log log by resorting to known integer handling data structures

a formula to construct all involutions in riordan matrix groups abstract we get a formula for all finite riordan involutions using the natural approximation induced by the inverse limit construction we also describe all infinite riordan matrices which are involutions in the riordan group finally we get some consequences

double parameter recurrences for polynomials in bi infinite riordan matrices and some derived identities abstract we present some double parameter recurrence relations for families of polynomials in bi infinite riordan matrices from this point of view we treat some riordan families of generalized fibonacci and lucas polynomials we use these to obtain some identities related mainly with binomial coefficients and fibonacci numbers

matrix characterizations of riordan arrays abstract here we discuss two matrix characterizations of riordan arrays p matrix characterization and a matrix characterization p matrix is an extension of the stieltjes matrix defined in 28 and the production matrix defined in 8 by modifying the marked succession rule introduced in 21 a combinatorial interpretation of the p matrix is given the p matrix characterizations of some subgroups of riordan group are presented which are used to find some algebraic structures of the subgroups we also give the p matrix characterizations of the inverse of a riordan array and the product of two riordan arrays a matrix characterization is defined in 20 and it is proved to be a useful tool for a riordan array while on the other side the a sequence characterization is very complex sometimes by using the fundamental theorem of riordan arrays a method of construction of a matrix characterizations from riordan arrays is given the converse process is also discussed several examples and applications of two matrix characterizations are presented

algebraic aspects of some riordan arrays related to binary words avoiding a pattern we consider some riordan arrays related to binary words avoiding a pattern p which can be easily studied by means of an a matrix rather than their a sequence both concepts allow us to define every element as a linear combination of other elements in the array the a sequence is unique and corresponds to a linear dependence from the previous row the a matrix is not unique and corresponds to a linear dependence from several previous rows however for the problems considered in the present paper we show that the a matrix approach is more convenient we provide explicit algebraic generating functions for these riordan arrays and obtain many statistics on the corresponding languages we thus obtain a deeper insight of the languages l p of binary words avoiding p having a number of 0s less or equal to the number of 1s

riordan arrays and d orthogonality abstract using the theory of the riordan group and d orthogonal polynomials we shall show that sequences of d orthogonal polynomials can be also generated by the riordan group we interpret some families of d orthogonal polynomials within the framework of the riordan group

an application of riordan arrays to the transient analysis of m m 1 queues abstract riordan arrays are used extensively in many contexts as a combinatorial tool for solving enumeration problems such as lattice path counting problems in this paper we first show how riordan array method can be applied to the transient analysis of m m 1 queue with zero customer at the initial point second we extend this method to the generalized riordan array with multiple support functions in order to deal with the transient analysis of m m 1 queue with non zero customers at the initial point numerical examples are also given to show how easy and quick the transient probability obtained from the riordan method can be computed

riordan graphs i structural properties abstract in this paper we use the theory of riordan matrices to introduce the notion of a riordan graph the riordan graphs are a far reaching generalization of the well known and well studied pascal graphs and toeplitz graphs and also some other families of graphs the riordan graphs are proved to have a number of interesting fractal properties which can be useful in creating computer networks with certain desirable features or in obtaining useful information when designing algorithms to compute values of graph invariants the main focus in this paper is the study of structural properties of families of riordan graphs obtained from infinite riordan graphs which includes a fundamental decomposition theorem and certain conditions on riordan graphs to have an eulerian trail cycle or a hamiltonian cycle we will study spectral properties of the riordan graphs in a follow up paper

steady state of stochastic sandpile models we study the steady state of the abelian sandpile models with stochastic toppling rules the particle addition operators commute with each other but in general these operators need not be diagonalizable we use their abelian algebra to determine their eigenvalues and the jordan block structure these are then used to determine the probability of different configurations in the steady state we illustrate this procedure by explicitly determining the numerically exact steady state for a one dimensional example for systems of size 12 and also study the density profile in the steady state

a symbolic treatment of riordan arrays abstract we approach riordan arrays and their generalizations via umbral symbolic methods this new approach allows us to derive fundamental aspects of the theory of riordan arrays as immediate consequences of the umbral version of the classical abel s identity for polynomials in particular we obtain a novel non recursive formula for riordan arrays and derive from this new formula some known recurrences and a new recurrence relation for riordan arrays

exact solution of the totally asymmetric oslo model recently it has been found pruessner g and jensen h j 2003 phys rev lett 91 244303 that a totally asymmetric variant of the oslo model christensen k et al 1996 phys rev lett 77 107 represents the entire universality class of the oslo model with anisotropy the totally asymmetric model can be solved without scaling assumptions by finding recursively the eigenvectors of the markov matrix which can be suitably modified to produce the moment generating function of the relevant observable this method should be applicable to many other stochastic processes

characteristics of deterministic and stochastic sandpile models in a rotational sandpile model rotational constraint representing a local external bias generally has a nontrivial effect on the critical behavior of lattice statistical models in equilibrium critical phenomena in order to study the effect of rotational bias in an out of equilibrium situation like self organized criticality a two state quasideterministic rotational sandpile model is developed here imposing rotational constraint on the flow of sand grains an extended set of critical exponents are estimated to characterize the avalanche properties at the nonequilibrium steady state of the model the probability distribution functions are found to obey usual finite size scaling supported by negative time autocorrelation between the toppling waves the model exhibits characteristics of both deterministic and stochastic sandpile models

half of a riordan array and restricted lattice paths abstract for an infinite lower triangular matrix g g n k n k 0 we define the half of g to be the infinite lower triangular matrix h h n k n k 0 such that h n k g 2 n k n for all n k 0 in this paper we will show that if g g n k n k 0 is a riordan array then its half h h n k n k 0 is also a riordan array and we obtain new combinatorial interpretations for some riordan arrays in terms of weighted lattice paths

some inverse limit approaches to the riordan group abstract using the inverse limit tool we obtain the riordan group in its infinite and bi infinite representations from groups of finite riordan matrices we define two different reflections for bi infinite riordan matrices employing these definitions we give answers to the problems d d and d d in the riordan group so we describe the self complementary and self dual riordan matrices

elements of finite order in the riordan group and their eigenvectors abstract we consider elements of finite order in the riordan group r over a field f of characteristic 0 we solve for all n 2 two foundational questions posed by l shapiro 2001 for the case n 2 involutions given f x of finite compositional order n theorem 1 gives a formula for g x showing that those g x which make g x f x r a riordan element of order n are precisely those for which the matrix g x f x has an eigenvector h 0 h 1 t with h 0 0 theorem 2 classifies finite order riordan group elements up to conjugation in r and gives a normal form for finite order riordan arrays under similarity this leads to theorem 3 a formula for all eigenvectors of finite order riordan arrays and results in interesting combinatorial identities we relate our work to 2008 and 2013 papers of cheon and kim which motivated this paper and we solve their open question this circle of ideas closes with a new proof of c marshall s theorem which finds the unique f x given bi invertible g x such that g x f x is an involution

recurrence relations for polynomial sequences via riordan matrices we give recurrence relations for any family of generalized appell polynomials unifying so some known recurrences for many classical sequences of polynomials our main tool to get our goal is the riordan group we use the product of riordan matrices to interpret some relationships between different polynomial families moreover using the hadamard product of series we get a general recurrence relation for the polynomial sequences associated to the so called generalized umbral calculus

hidden word statistics we consider the sequence comparison problem also known as hidden pattern problem where one searches for a given subsequence in a text rather than a string understood as a sequence of consecutive symbols a characteristic parameter is the number of occurrences of a given pattern w of length m as a subsequence in a random text of length n generated by a memoryless source spacings between letters of the pattern may either be constrained or not in order to define valid occurrences we determine the mean and the variance of the number of occurrences and establish a gaussian limit law and large deviations these results are obtained via combinatorics on words formal language techniques and methods of analytic combinatorics based on generating functions the motivations to study this problem come from an attempt at finding a reliable threshold for intrusion detections from textual data processing applications and from molecular biology

palindromes and pseudo involution multiplication abstract a riordan array g f is called a pseudo involution if g f m or equivalently m g f where m 1 z is an involution this paper presents a palindromic property of pseudo involutions which seems both novel and useful if a and b are both pseudo involutions then so is the triple product aba with this it follows that if a b c are pseudo involutions so is any palindromic word using these symbols

sandpile on directed small world networks we numerically investigate the avalanche dynamics of the bak tang wiesenfeld sandpile model on directed small world networks we find that the avalanche size and duration distribution follow a power law for all rewiring probabilities p specially we find that approaching the thermodynamic limit l the values of critical exponents do not depend on p and are consistent with the mean field solution in euclidean space for any p 0 in addition we measure the dynamic exponent in the relation between avalanche size and avalanche duration and find that the values of the dynamic exponents are also consistent with the mean field values for any p 0

rare events and breakdown of simple scaling in the abelian sandpile model due to intermittency and conservation the abelian sandpile in two dimensions obeys multifractal rather than finite size scaling in the thermodynamic limit a vanishingly small fraction of large avalanches dominates the statistics and a constant gap scaling is recovered in higher moments of the toppling distribution thus rare events shape most of the scaling pattern and preserve a meaning for effective exponents which can be determined on the basis of numerical and exact results

a unified approach for the catalan matrices by using riordan arrays abstract we study generalized catalan matrices based on the riordan array and fuss catalan numbers a unified combinatorial interpretation for the entries of the generalized catalan matrices is presented by means of m dyck paths some properties and examples of the generalized catalan matrices are given including a new convolution formula for the generalized catalan numbers finally we present applications of generalized catalan matrices to the problems in counting the hill free and lower peak free m dyck paths

fuss catalan matrices their weighted sums and stabilizer subgroups of the riordan group abstract in this paper we present the riordan arrays called fuss catalan matrices which are constructed by the convolutions of the generating functions of the fuss catalan numbers we also discuss weighted sums of the fuss catalan matrices using such matrices as transformations of recursive sequences and their connection with stabilizer subgroups of the riordan group

a combinatorial equivalence relation for formal power series abstract in this paper we examine an equivalence relation on the set of formal power series with nonzero constant term this is done both in terms of functional equations and also by interlacing two concepts from riordan group theory the a sequence and the bell subgroup the best known example gives an equivalence class 1 z 1 1 z c z t z q z where c z t z and q z are generating functions of the catalan numbers ternary numbers and quaternary numbers respectively a power series for one member of an equivalence class can be transformed into power series for the rest of members in the equivalence class and interpretations in terms of weighted lattice paths can also be given

a characterization of graphs with rank 4 the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph with rank 4 this question has not yet been fully answered in the literature and only some partial results are known in this paper we resolve this question by completely characterizing graphs g whose adjacency matrix has rank 4

groups of generalized pascal matrices abstract we present a new approach to the study of generalized pascal matrices that yields general results about group structures and explicit formulas for powers and inverses of the pascal matrices shows how the groups of pascal matrices are related to groups of 2 2 matrices and clarifies the way in which symmetries and duality relate to the algebraic aspects of the theory we consider bilaterally infinite generalized pascal matrices obtained as the matrix representations of certain linear operators on spaces of formal laurent series their column generating functions are simple rational functions closely related to the linear fractional maps of the complex plane we obtain explicit expressions for the inverses and the powers of the pascal matrices that generalize most of the analogous results in the literature

identities induced by riordan arrays historically there exist two versions of the riordan array concept the older one better known as recursive matrix consists of bi infinite matrices d n k n k is an element of z k n implies d n k 0 deals with formal laurent series and has been mainly used to study algebraic properties of such matrices the more recent version consists of infinite lower triangular arrays d n k n k is an element of n k n implies d n k 0 deals with formal power series and has been used to study combinatorial problems here we show that every riordan array induces two characteristic combinatorial sums in three parameters n k m is an element of z these parameters can he specialized and generate an indefinite number of other combinatorial identities which are valid in the hi infinite realm of recursive matrices

universality classes in isotropic abelian and non abelian sandpile models universality in isotropic abelian and non abelian sandpile models is examined using extensive numerical simulations to characterize the critical behavior we employ an extended set of critical exponents geometric features of the avalanches as well as scaling functions describing the time evolution of average quantities such as the area and size during the avalanche comparing between the abelian bak tang wiesenfeld model p bak c tang and k wiensenfeld phys rev lett 59 381 1987 and the non abelian models introduced by manna s s manna j phys a 24 l363 1991 and zhang y c zhang phys rev lett 63 470 1989 we find strong indications that each one of these models belongs to a distinct universality class

riordan arrays and related polynomial sequences abstract in this paper we generalize the results due to luzon and moron and present a characterization of the generalized riordan arrays using this characterization we establish a recurrence for the generalized sheffer sequences and study some special types of polynomial sequences including the generalized lucas u sequence and v sequence and the generalized humbert polynomial sequence in particular by appealing to the factorizations of riordan arrays the umbral composition of polynomial sequences and the hadamard product of series we establish various relations among these polynomial sequences thus we give a unified approach to some well known polynomial sequences appeared in combinatorics and the theory of special functions

combinatorics of a generalized narayana identity abstract the narayana identity is a well known formula that expresses the classical catalan numbers as sums of the ordinary narayana numbers in this paper we generalize the narayana identity to a family of riordan arrays including the array of ballot numbers the classical catalan triangle and several generalized catalan triangles recently studied a combinatorial description based on non crossing partitions is given for this identity for the column recursive rule and for the sheffer sequence associated with any array of the family

recursive matrices and umbral calculus two major objectives can be seen to guide much recent work in enumeration 1 to single out a limited variety of recurrences for numerical sequences which will encompass counting problems of wide enough type 2 to recover from empirical data an underlying set theoretic structure which would reveal the source of the given recursion we are here concerned with the first of these objectives though the eventual understanding of the second is tacitly present if only as a goal we noticed the coincidence of several computations which similar as they are in retrospect had failed to realize their kinship on leafing through the unique assembly of recursively solvable combinatorial problems in comtet s and sloane s invaluable collections one is struck by the repeated occurrence of one and the same kind of double recursion more strikingly the same recursion is seen to occur in the polynomial sequences of the umbra1 calculus of roman and rota see 25 everywhere the lagrange inversion formula for power series plays a pivotal role much work is nowadays going into the unraveling of the everdeeper layers of combinatorial significance of this formula both in the ordinary case and in its as yet partially worked out noncommutative and qanalogs andrews foata garsia gessel joni raney reiner schtitzenberger to name but a few whatever their origins the identities abutting lagrange inversion are expressed by integers alone this suggests not only a hidden set theoretic layer but a characteristic free generalization as well this generalization is the central theme of our work we define a monoid of infinite matrices recursive matrices for short the entries of these matrices give the sought out recursion for example that for coefficients of binomial and sheffer polynomials and factor sequences as well as that of the special sequences recently introduced by roman in 26 546 002 l 8693 82 040546 28 02 00 0

per spectral characterizations of graphs with extremal per nullity abstract a graph g is said to be determined by its permanental spectrum if any graph having the same permanental spectrum as g is isomorphic to g in this paper we introduce the permanental nullity of a graph the multiplicity of the number zero in the permanental spectrum of a graph to study graphs determined by their permanental spectra first we determine all graphs of order n whose permanental nullities are n 2 n 3 n 4 and n 5 respectively then we show that all graphs with the permanental nullity n 2 n 3 or n 5 and all non bipartite graphs with the permanental nullity n 4 are determined by their permanental spectra in particular we prove that the complete bipartite graphs are determined by their permanental spectra

notes on the total positivity of riordan arrays abstract let r d t h t be a riordan array we show that if both d t and h t are polya frequency formal power series then r is totally positive

q riordan representation abstract the aim of this study is to establish further theory of q riordan representation we emphasize that any q matrices can be obtained by q riordan representation then we analyse the structure of the entries in the matrix product of general q riordan matrices

dual operators and lagrange inversion in several variables the original lagrange inversion formula which gives explicitly the inverse under composition of a formal series was obtained by lagrange in 1770 through formal computations involving logarithms of inlmite products see lagrange 121 there exists an extensive literature on various versions of the inversion formula obtained by algebraic analytic and combinatorial methods comtet 3 gives a survey and an ample bibliography during the last ten years there have appeared several papers on generalizations and new proofs of the inversion formula in 1974 abhyankar obtained an inversion formula using algebraic methods see bass connell and wright 11 several authors have related lagrange inversion with rota s finite operator calculus 151 for example joni 8 garsia and joni 4 hofhauer 7 and roman and rota 14 joyal and labelle used a combinatorial theory of formal series to obtain inversion formulas see 9 l i and viskov 161 used ideas from lie algebra theory to prove formulas similar to abhyankar s in the present paper we use operator methods to obtain inversion formulas in several variables that generalize the formulas of abhyankar joni and viskov we get our results studying an algebra with involution of linear operators on the algebra of formal laurent series in several indeterminates we also show that by means of an anti isomorphism of operator algebras which we call the operator bore1 transform the study of finite operator calculus can be reduced to the study of certain groups of operators on the algebra of formal laurent series our approach may be considered as an algebraic analogue of the complex variable methods used by good 5 to obtain the lagrange inversion formula as a consequence of the change of variables theorem for the

finite and infinite dimensional lie group structures on riordan groups abstract we introduce a frechet lie group structure on the riordan group we give a description of the corresponding lie algebra as a vector space of infinite lower triangular matrices we describe a natural linear action induced on the frechet space k n by any element in the lie algebra we relate this to a certain family of bivariate linear partial differential equations we obtain the solutions of such equations using one parameter groups in the riordan group we show how a particular semidirect product decomposition in the riordan group is reflected in the lie algebra we study the stabilizer of a formal power series under the action induced by riordan matrices we get stabilizers in the fraction field k x using bi infinite representations we provide some examples the main tool to get our results is the paper 18 where the riordan group was described using inverse sequences of groups of finite matrices

exact results for the directed abelian sandpile models we define the directed abelian sandpile models by introducing a parameter c representing the degree of anisotropy in the avalanche processes where c 1 is for the isotropic case we calculate some quantities characterizing the self organized critical states on the one and two dimensional lattices in particular we obtain the expected number of topplings per added particle t which shows the dependence on the lattice size l as lx for large l we show that the critical exponent x does not depend on the dimensionality d at least for d 1 and 2 and that when any anisotropy is included in the system x 1 while x 2 in the isotropic system this result gives a rigorous proof of the conjecture by kadanoff et al 1989 a 39 6524 37 that the anisotropy will distinguish different universality classes we introduce a new critical exponent defined by x liml t l with as for both in d 1 and 2 we obtain 1

a general umbral calculus in infinitely many variables on etudie des suites de fonctions factorielles satisfaisant des analogues des recurrences binomiales et de sheffer et on montre comment un calcul operatoriel ressemblant au calcul umbral peut etre construit dessus

a determinantal approach to sheffer sequences abstract in this paper by using the theory of riordan arrays and the relations between sheffer sequences and riordan arrays we give a determinantal definition for sheffer sequences based on this new definition some general properties of sheffer sequences are reproved and the determinantal representations of some well known sheffer sequences are presented

the abelian sandpile and related models abstract the abelian sandpile model is the simplest analytically tractable model of self organized criticality this paper presents a brief review of known results about the model the abelian group structure of the algebra of operators allows an exact calculation of many of its properties in particular when there is a preferred direction one can calculate all the critical exponents characterizing the distribution of avalanche sizes in all dimensions for the undirected case the model is related to q 0 state potts model this enables exact calculation of some exponents in two dimensions and there are some conjectures about others we also discuss a generalization of the model to a network of communicating reactive processors this includes sandpile models with stochastic toppling rules as a special case we also consider a nondashabelian stochastic variant which lies in a different universality class related to directed percolation

bijections and the riordan group one of the cornerstone ideas in mathematics is to take a problem and to look at it in a bigger space in this paper we examine combinatorial sequences in the context of the riordan group various subgroups of the riordan group each give us a different view of the original sequence in many cases this leads to both a combinatorial interpretation and to eco rewriting rules in this paper we will concentrate on just four of the subgroups of the riordan group to demonstrate some of the possibilities of this approach

computing the permanental polynomials of graphs let m be an nn matrix with entries mij i j 1 2 n the permanent of m is defined to be per m i 1nmi i where the sum is taken over all permutations of 1 2 n the permanental polynomial of m is defined by per xinm where in is the identity matrix of size n in this paper we give recursive formulas for computing permanental polynomials of the laplacian matrix and the signless laplacian matrix of a graph respectively

structural properties of riordan matrices and extending the matrices abstract we consider an infinite lower triangular matrix l l n k n k n 0 and a sequence n n n 0 called the a b sequence such that every element l n 1 k 1 except lying in column 0 can be expressed as l n 1 k 1 i 0 n k m i l n ai k bi 0 0 where a and b are integers with a b m 0 and b 0 this concept generalizes the a sequence of a riordan matrix as a result we explore several structural properties of riordan matrices by means of a b sequences in particular if b

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

exactly solved model of self organized critical phenomena we define a variant of the model of bak tang and wiesenfeld of self organized critial behavior by introducing a preferred direction we characterize the critical state and by establishing equivalence to a voter model determine the critical exponents exactly in arbitrary dimension d the upper critical dimension for this model is three in two dimensions the model is equivalent to an earlier solved special case of directed percolation

self organized branching processes avalanche models with dissipation we explore in the mean field approximation robustness with respect to dissipation of self organized criticality in sandpile models to this end we generalize a recently introduced self organized branching process and show that the model self organizes not into a critical state but rather into a subcritical state when dissipation is present the dynamical fixed point does not coincide with the critical point thus the level of dissipation acts as a relevant parameter in the renormalization group sense we study the model numerically and compute analytically the critical exponents for the avalanche size and lifetime distributions and the scaling exponents for the corresponding cutoffs s1063 651x 96 08009 9

riordan matrices in the reciprocation of quadratic polynomials we iterate contractive one degree polynomials with coefficients in the ring k x of formal power series to calculate the reciprocal in k x of a quadratic polynomial doing this we meet the structure of riordan array we interpret certain changes of variable as a riordan array we finish the paper by using our techniques to find new ways to get known formulas for the sum of powers of natural numbers involving stirling and eulerian numbers

recurrence relations for the sheffer sequences abstract in this paper using the production matrix of an exponential riordan array g t f t we give a recurrence relation for the sheffer sequence for the ordered pair g t f t we also develop a new determinant representation for the general term of the sheffer sequence as applications determinant expressions for some classical sheffer polynomial sequences are derived

the relevant prefixes of coloured motzkin walks an average case analysis in this paper we study some relevant prefixes of coloured motzkin walks otherwise called coloured motzkin words in these walks the three kinds of step can have a b and c colours respectively in particular when a b c 1 we have the classical motzkin walks while for a c 1 and b 0 we find the well known dyck walks by using the concept of riordan arrays and probability generating functions we find the average length of the relevant prefix in a walk of length n and the corresponding variance in terms of a b and c this result is interesting from a combinatorial point of view and also provides an average case analysis of algorithms related to the problem of ranking and generating uniformly at random the coloured motzkin words

renormalization group for directed sandpile models following the lead of earlier researchers we construct a real space renormalization group map for a class of self organized critical models we then test a series of quantitative predictions implied by this map against direct simulations of various models the theory passes four tests but apparently fails a fifth we also find that the map has an interesting nongeneric property suggesting a deeper structure to the theory than previously appreciated s0031 9007 98 06840 9

universality in sandpiles we perform extensive numerical simulations of different versions of the sandpile model we find that previous claims about universality classes are unfounded since the method previously employed to analyze the data suffered from a systematic bias we identify the correct scaling behavior and provide evidences suggesting that sandpiles with stochastic and deterministic toppling rules belong to the same universality class

logarithmic corrections of the avalanche distributions of sandpile models at the upper critical dimension we study numerically the dynamical properties of the btw model on a square lattice for various dimensions the aim of this investigation is to determine the value of the upper critical dimension where the avalanche distributions are characterized by the mean field exponents our results are consistent with the assumption that the scaling behavior of the four dimensional btw model is characterized by the mean field exponents with additional logarithmic corrections we benefit in our analysis from the exact solution of the directed btw model at the upper critical dimension which allows to derive how logarithmic corrections affect the scaling behavior at the upper critical dimension similar logarithmic corrections forms fit the numerical data for the four dimensional btw model strongly suggesting that the value of the upper critical dimension is four

how self organized criticality works a unified mean field picture we present a unified dynamical mean field theory based on the single site approximation to the master equation for stochastic self organized critical models in particular we analyze in detail the properties of sandpile and forest fire ff models in analogy with other nonequilibrium critical phenomena we identify an order parameter with the density of active sites and control parameters with the driving rates depending on the values of the control parameters the system is shown to reach a subcritical absorbing or supercritical active stationary state criticality is analyzed in terms of the singularities of the zero field susceptibility in the limit of vanishing control parameters the stationary state displays scaling characteristics of self organized criticality soc we show that this limit corresponds to the breakdown of space time locality in the dynamical rules of the models we define a complete set of critical exponents describing the scaling of order parameter response functions susceptibility and correlation length in the subcritical and supercritical states in the subcritical state the response of the system to small perturbations takes place in avalanches we analyze their scaling behavior in relation with branching processes in sandpile models because of conservation laws a critical exponents subset displays mean field values ensuremath nu frac 1 2 and ensuremath gamma 1 in any dimensions we treat bulk and boundary dissipation and introduce a critical exponent relating dissipation and finite size effects we present numerical simulations that confirm our results in the case of the forest fire model our approach can distinguish between different regimes soc ff and deterministic ff studied in the literature and determine the full spectrum of critical exponents