Donatella Merlini

• Professor (Ph.D)
• Professor (Associate) at University of Florence

As already describedSpecial sequences in the previous chapters, the concept of a Riordan array was introduced in 1991 by Shapiro, Getu, Woan and Woodson [21], with the aim of defining a class of infinite lower triangular arrays with properties analogous to those of the Pascal triangle. This concept was subsequently studied by Sprugnoli [22] in the...

In this chapter, we give an overview of the links between Riordan arrays and orthogonal polynomials, and then we study some specialized areas including classical and semi-classical orthogonal polynomials defined by Riordan arrays, orthogonal polynomials that can be described as the moment sequences of Riordan arrays, applications of exponential Rio...

The previous chapters have shown that the theory of Riordan arrays is a powerful tool for studying combinatorial sums and special polynomial and number sequences. One of the well-known classes of polynomial sequences is the class of Sheffer sequences, including many important sequences such as Bernoulli polynomials, Euler polynomials, Abel polynomi...

Traditional methods used for solving combinatorial sums (see, e.g., J. Riordan [18] or L. Comtet [1]) are used in R. L. Graham, D. E. Knuth, and O. Patashnik [8], where it is shown how to use the rules of binomial coefficients, Stirling numbers, and so on, for computing combinatorial sums. G. P. Egorychev [3] developed the method known as integral...

The group of Riordan arrays was introduced in 1991 by Shapiro, Getu, Woan, and Woodson [15], with the aim of defining a class of infinite lower triangular arrays with properties analogous to those of the Pascal triangle. A previous generalization of the Pascal, Catalan, and Motzkin triangles can be found in Rogers [13] who introduces the concept of...

We consider two groups, F0={g∈F[[z]]∣g(0)≠0} under multiplication, and F1=zF0 under composition where F is the real field R or complex field C. As observed in Sect. 3.3, it is known that the Riordan group R is isomorphic to the semidirect product F0⋊F1. It may be viewed as a group extension of F1 by F0. In this chapter, we develop the group of thre...

The Riordan group consisting of proper Riordan arrays shows up naturally in a variety of combinatorial settings. In this chapter, we define the q-analog of a Riordan array called a q-Riordan array and denoted as (g,f)q. It is defined by using a pair of Eulerian generating functions g, f of the form ∑n≥0anzn[n]q!where [n]q!=1(1+q)(1+q+q2)⋯(1+q+⋯+qn-...

Generating functions have emerged as one of the most popular approaches to combinatorial problems, above all to problems arising in the analysis of algorithms (see, for example, D. E. Knuth [8] and R. Sedgewick and Ph. Flajolet [11] and Ph. Flajolet and R. Sedgewick [3]). A clear exposition of this concept is given in three books, namely, those of...

In this paper we consider courses of a Computer Science degree in an Italian university from the year 2011 up to 2020. For each course, we know the number of exams taken by students during a given calendar year and the corresponding average grade; we also know the average normalized value of the result obtained in the entrance test and the distribu...

In this paper, by using Riordan arrays and a particular model of lattice paths, we are able to generalize in several ways an identity proposed by Lou Shapiro by giving both an algebraic and a combinatorial proof. The identities studied in this paper allow us to move from an arithmetic progression, and other C-finite sequences, to a geometric progre...

Students of the University of Florence (Italy), before taking an exam, are required to assess different aspects related to the course organization and to the teaching. The data concerning the evaluation of the courses of the Computer Science Program from 2001/2002 to 2007/2008 academic years were collected and linked to the results of students: the...

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 and in the Riordan group. So, we describe the self-complementary and self...

This paper presents a data mining methodology to analyze the careers of University graduated students. We present different approaches based on clustering and sequential patterns techniques in order to identify strategies for improving the performance of students and the scheduling of exams. We introduce an ideal career as the career of an ideal st...

Recently, the concept of the complementary array of a Riordan array (or recursive matrix) has been introduced. Here we generalize the concept and distinguish between dual and complementary arrays. We show a number of properties of these arrays, how they are computed and their relation with inversion. Finally, we use them to find explicit formulas f...

Historically, there exist two versions of the Riordan array concept. The older one (better known as recursive matrix) consists of bi-infinite matrices (dn,k)n,k∈Z(dn,k)n,k∈Z (k>nk>n implies dn,k=0dn,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 inf...

In this paper we study the enumeration and the construction of particular binary words avoiding the pattern 1j+10j. By means of the theory of Riordan arrays, we solve the enumeration problem and we give a particular succession rule, called jumping and marked succession rule, which describes the growth of such words according to their number of ones...

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....

In this paper we study the enumeration and the construction of particular binary words avoiding the pattern $1^{j+1}0^j$. By means of the theory of Riordan arrays, we solve the enumeration problem and we give a particular succession rule, called jumping and marked succession rule, which describes the growth of such words according to their number o...

In this paper, we study the stationary node distribution of a variation of the Random Waypoint mobility model, in which nodes move in a smooth way following one randomly chosen Manhattan path connecting two points. We provide analytical results for the spatial node stationary distribution of this model. As an application, we exploit this result to...

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 α,β and γ colours, respectively. In particular, when α=β=γ=1 we have the classical Motzkin walks while for α=γ=1 and β=0 we find the well-known Dyck walks. By using the concept of Riorda...

In this paper we present the theory of implicit Riordan arrays, that is, Riordan arrays which require the application of the Lagrange Inversion Formula to be dealt with. We show several examples in which our approach gives explicit results, both in finding closed expressions for sums and, especially, in solving classes of combinatorial sum inversio...

We study some combinatorial properties of Tetris-like games by using Schutzenberger methodology and probability generating functions. We prove that every Tetris-like game is equivalent to a finite state automaton and propose a straight Coward algorithm to transform a Tetris-like game into its corresponding automaton. In this way, we can study the a...

We find the generating function counting the total internal path length of any proper generating tree. This function is expressed in terms of the functions (d(t),h(t)) defining the associated proper Riordan array. This result is important in the theory of Riordan arrays and has several combinatorial interpretations.

We introduce a generalization of generating trees named LevelGenerating Trees and study the connection between these structuresand proper {sc Riordan} arrays, deriving a theorem that, under suitableconditions, associates a {sc Riordan} array to a Level Generating Treeand vice versa. We illustrate our main results by several examplesconcerning class...

Say an integer n is exceptional if the maximum Stirling number of the second kind S(n,k) occurs for two (of necessity consecutive) values of k. We prove that the number of exceptional integers less than or equal to x is O(x 1/2+ε ), for any ε>0. We derive a similar result for partitions of n into exactly k integers.

We study two remarkable identities of Andrews relating Fibonacci numbers and binomial coefficients in terms of generating functions and Riordan arrays. We thus give a deeper insight to the problem and find several new identities involving many other well-known sequences.

We consider the uniform generation of random derangements, i.e., permuta-tions without any fixed point. By using a rejection algorithm, we improve the straight-forward method of generating a random permutation until a de-rangement is obtained. This and our procedure are both linear with respect to the number of calls to the random generator, but we...

We study some combinatorial properties related to the problem of tablature for stringed instruments. First, we describe the problem in a formal way and prove that it is equivalent to a finite state automaton. We define the concepts of distance between two chords and tablature complexity in order to study the problem of tablature in terms of music p...

We study the relation between binary words excluding a pattern and proper Riordan arrays. In particular, we prove necessary and sufficient conditions under which the number of words counted with respect to the number of zeroes and ones bits are related to proper Riordan arrays. We also give formulas for computing the generating functions (d(x),h(x)...

The paper gives an account of the "method of coefficients" due to G. P. Egorychev. The method is used, often without any explicit reference, in the practice of formal power series and generating functions, both in combinatorics and in the analysis of algorithms. Here we show how we can start with a restricted series of general rules and proceed to...

In this paper we present the theory of implicit Riordan arrays, that is, Riordan arrays which require the application of the Lagrange Inversion Formula to be dealt with. We show several examples in which our approach gives explicit results in solving classes of combinatorial sum inversions.

We study many properties of Cauchy numbers in terms of generating functions and Riordan arrays and find several new identities relating these numbers with Stirling, Bernoulli and harmonic numbers. We also reconsider the Laplace summation formula showing some applications involving the Cauchy numbers.

We investigate the algebraic rules for functionally inverting a Riordan array given by means of two analytic functions. In this way, we find an extension of the Lagrange Inversion Formula and we apply it to some combinatorial problems on simple coloured walks. For some of these problems we give both an algebraic and a combinatorial proof.

The aim of the present paper is to show how the Lagrange Inversion Formula (LIF) can be applied in a straight-forward way i) to find the generating function of many combinatorial sequences, ii) to extract the coefficients of a formal power series, iii) to compute combinatorial sums, and iv) to perform the inversion of combinatorial identities. Part...

International audience It has become customary to prove binomial identities by means of the method for automated proofs as developed by Petkovšek, Wilf and Zeilberger. In this paper, we wish to emphasize the role of "human'' and constructive proofs in contrast with the somewhat lazy attitude of relaying on "automated'' proofs. As a meaningful examp...

We consider the transformation proposed by Akiyama and Tanigawa to obtain the Bernoulli numbers and extend it to other important sequences. We also show its connec-tion to two particular Riordan Arrays, which allow us to prove a number of combinatorial sums related to the transformation.

We introduce a model based on some combinatorial objects, which we call 1-histograms, to study the behaviour of devices like printers and use the combinatorial properties of these objects to study some important distributions such as the waiting time for a job and the length of the device queue. This study is based on an important relation between...

We study many properties of compositions of integers by using the symbolic method, multivariate generating functions and Riordan arrays. In particular, we study palin-dromic compositions with respect to various parameters and present a bijection between walks, compositions and bit strings. The results obtained for compositions can thus be exported...

Whereas walks on N with a finite set of jumps were the subject of numerous studies, walks with an infinite number of jumps remain quite rarely studied, at least from a combinatorial point of view. A reason is that even for relatively well structured models, the classical approach with context-free grammars fails as we deal with rewriting rules over...

We derive asymptotic expansions for the Stirling numbers of real arguments as defined by Flajolet and Prodinger. We also generalize certain classical identities for Stirling numbers with integral arguments to real or complex arguments.

We study some lattice paths related to the concept of generating trees. When the matrix associated to this kind of trees is a Riordan array D d t h t , we are able to find the generating function for the total area below these paths expressed in terms of the functions d t and h t

International audience We study some lattice paths related to the concept ofgenerating trees. When the matrix associated to this kind of trees is a Riordan array $D=(d(t),h(t))$, we are able to find the generating function for the total area below these paths expressed in terms of the functions $d(t)$ and $h(t)$.

n=k 0 1 2 3 4 0 1 1 0 1 2 1 0 1 3 0 2 0 1 4 2 0 3 0 1 = Henri Delannoy triangular chessboards (1886). = Riordan arrays [7,10,11]. Walks on Z with an in nite set of negative jumps Consider a sequence (e i (k)) ia (for a given integer a > 0) of polynomials assuming nonnegative integers values: e k 1 (k) e 0 (k) e a (k) The exponent e i (k) is the mul...

Mallows and Shapiro, (J. Integer Sequences2 (1999)) have recently considered what they dubbed the problem of balls on the lawn. Our object is to explore a natural generalization, the s-tennis ball problem, which reduces to that considered by Mallows and Shapiro in the case s=2. We show how this generalization is connected with s-ary trees, and empl...

We consider the well-known problem of how to arrange coins in a fountain and generalize it, by using combinatorial objects known as staircase polyominoes or p-histograms. We show how it is possible to obtain an asymptotic formula approximating the number of p-histograms of area n, as n→∞.

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 st...

We study some statistics related to Dyck paths, whose explicit formulas are obtained by means of the Lagrange Inversion Theorem. There are five such statistics and one of them is well-known and owed to Narayana. The most interesting of the other four statistics is related to Euler's trinomial coefficients and to Motzkin numbers: we perform a study...

We give an alternative proof of an identity that appeared recently in Integers. By using the concept of Riordan arrays we obtain a short, elementary proof.

Our object is to explore "the s-tennis ball problem" (at each turn s balls are available and we play with one ball at a time). This is a natural generalization of the case s = 2 considered by Mallows and Shapiro. We show how this generalization is connected with s-ary trees and employ the notion of generating trees to obtain a solution expressed in...

We find an algebraic structure for a subclass of generating trees by introducing the concept of marked generating trees. In these kind of trees, labels can be marked or non marked and the count relative to a certain label at a certain level is given by the difference between the number of non marked and marked labels. The algebraic structure corres...

We consider a new method to retrieve keys in a static table. The keys of the table are stored in such a way that a binary search can be performed more efficently. An analysis of the method is performed and empirical evidence is given that it actually works.

We present a C++ program which implements an algorithm able to solve strip tiling problems. The program transforms a specified problem into a regular grammar and then finds the generating function counting the number of different ways the strip can be tiled with the given pieces. Some examples and applications are illustrated.

We propose a method to solve a class of terminating hypergeometric series 2F 1 by using generating functions and the Lagrange inversion formula. Altough the approach is not as general as the WZ-method, nonetheless it is rather effective and succeeds in solving sums that cannot be treated by similar methods, as the ones proposed for the solutions of...

We define the Stirling numbers for complex values and obtain extensions of certain identities involving these numbers. We also show that the generalization is a natural one for proving unimodality and monotonicity results for these numbers. The definition is based on the Cauchy integral formula and can be used for many other combinatorial numbers....

. We use the "first passage decomposition" methodology to study the area between various kinds of underdiagonal lattice paths and the main diagonal. This area is important because it is connected to the number of inversions in permutations and to the internal path length in various types of trees. We obtain the generating functions for the total ar...

We characterize coloured Dyck and Schroder paths in both an algebraic and combinatorial way. In fact, we give algebraic and combinatorial proofs that, starting from the definition of such paths, we obtain a generating function and, from this, the corresponding recurrence. Finally, by using a generalization of a beautiful bijection of Sulanke [5], w...

We obtain a theorem that allows us to find the generating function for some combinatorial sums related to non proper Riordan arrays. This function can be used to obtain a closed form for the sum (or an asymptotic evaluation). We give several examples to illustrate some practical applications of the theorem. 1 Introduction A Riordan array D = fd n;k...

We present a method for obtaining asymptotics for the generic element of a twodimensional convolution matrix which includes Stirling numbers of both kinds and some other interesting combinatorial quantities. Asy2dim is a computer algebra package which implement this method. The current version of Asy2dim is written in Maple V.3 [2]. R'esum'e Nous p...

We use an algebraic approach to study the connection between generating trees and proper Riordan Arrays deriving a theorem that, under suitable conditions, associates a Riordan Array to a generating tree and vice versa. Thus, we can use results from the theory of Riordan Arrays to study properties of generating trees. In particular, we can find, in...

By using the method of Riordan arrays we find the asymptotic value of the expected storage utilization in a model of uniform B-tree-like structures proposed by Gupta and Srinivasan, thus solving the occupancy problem for that model. Keywords: analysis of algorithms, B-trees, Riordan arrays, storage utilization. 1 Introduction B-trees are one of the...

We study the problem of tiling a rectangular p Theta n-strip (p 2 N fixed, n 2 N) 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 co...

We give several new characterizations of Riordan Arrays, the most important of which is: if fd n;k g n;k2N is a lower triangular array whose generic element d n;k linearly depends on the elements in a well-defined though large area of the array, then fd n;k g n;k2N is Riordan. We also provide some applications of these characterizations to the latt...

We use some combinatorial methods to study underdiagonal paths (on the Z2 lattice) made up of unrestricted steps, i.e., ordered pairs of non-negative integers. We introduce an algorithm which automatically produces some counting generating functions for a large class of these paths. We also give an example of how we use these functions to obtain so...

We present the prototype of an algorithm animation system, seen as a tool for teaching basic algorithms in Computer Science. An important property of our system is the active role played by the student: he/she is allowed to interact with an algorithm in order to understand its behaviour (he/she can stop it, backtrack and rerun it) and can perform e...

In this paper we treat the static dictionary problem , very well known in computer science. It consists in storing a set S of m elements in the range [1 . . . n ] so that membership queries on S 's elements can be handled in O(1) time. It can be approached as a table compression problem in which a size n table has m ones and the other elements are...

The inversion of combinatorial sums is a fundamental problem in algebraic combinatorics. Some combinatorial sums, such as an = Σkdn,kbk, cannot be inverted in terms of the orthogonality relation because the infinite, lower triangular array P = {dn,k}'s diagonal elements are equal to zero (except d0,0). Despite this, we can find a left-inverse ̄P su...

In this paper, we study the spatial node stationary distribution and connectivity properties of two variations of the Random Waypoint (in short, RWP) mobility model. In particular, differently from the RWP mobility model that connects the source points to the destination ones by straight lines, our models make use of one of the two Manhattan paths...