Renzo Pinzani

• Professor Emeritus at University of Florence

Given a positive rational $q$, we consider Dyck paths having height at most two with some constraints on the number of consecutive peaks and consecutive valleys, depending on $q$. We introduce a general class of Dyck paths, called rational Dyck paths, and provide the associated generating function, according to their semilength, as well as the cons...

We consider Dyck paths having height at most two with some constraints on the number of consecutive valleys at height one which must be followed by a suitable number of valleys at height zero. We prove that they are enumerated by so-called Q-bonacci numbers (recently introduced by Kirgizov) which generalize the classical q-bonacci numbers in the ca...

We use Dyck paths having some restrictions in order to give a combinatorial interpretation for some famous number sequences. Starting from the Fibonacci numbers we show how the k -generalized Fibonacci numbers, the powers of 2, the Pell numbers, the k -generalized Pell numbers and the even-indexed Fibonacci numbers can be obtained by means of const...

Dyck paths having height at most $h$ and without valleys at height $h-1$ are combinatorially interpreted by means of 312-avoding permutations with some restrictions on their \emph{left-to-right maxima}. The results are obtained by analyzing a restriction of a well-known bijection between the sets of Dyck paths and 312-avoding permutations. We also...

We investigate the relationship between numerical semigroups and Dyck paths discovered by Bras-Amorós and de Mier. More specifically, we consider some classes of Dyck paths and characterize those paths giving rise to numerical semigroups.

Each positive increasing integer sequence {an}n≥0 can serve as a numeration system to represent each non-negative integer by means of suitable coefficient strings. We analyse the case of k-generalized Fibonacci sequences leading to the binary strings avoiding 1k. We prove a bijection between the set of such strings of length n and the set Sn+1(321,...

In From Fibonacci to Catalan permutations, Barcucci et al., 2006, a journey by pattern avoiding permutations from the Fibonacci to Catalan sequences is illustrated. In this paper we make a similar journey, but on Dyck paths.

Each positive increasing integer sequence $\{a_n\}_{n\geq 0}$ can serve as a numeration system to represent each non-negative integer by means of suitable coefficient strings. We analyse the case of $k$-generalized Fibonacci sequences leading to the binary strings avoiding $1^k$. We prove a bijection between the set %$F_n^{(k)}$ of strings of lengt...

We propose a method for the construction of sets of variable dimension strong non-overlapping matrices basing on any strong non-overlapping set of strings.

Each strictly increasing sequence of positive integers can be used to define a numeration system so that any non-negative integer can be represented by a suitable and unique string of digits. We consider sequences defined by a two termed linear recurrence with constant coefficients having some particular properties and investigate on the possibilit...

We propose a strong non-overlapping set of Dyck paths having variable length. First, we construct a set starting from an elevated Dyck path by cutting it in a specific point and inserting suitable Dyck paths (not too long...) in this cutting point. Then, we increase the cardinality of the set by replacing the first and the second factor of the orig...

We refer to positive lattice paths as to paths in the discrete plane constituted by different kinds of steps (north-east, east and south-east), starting from the origin and never going under the x-axis. They have been deeply studied both from a combinatorial and an algorithmic point of view. We propose some algorithms for the exhaustive generation...

Since some years, non-overlapping sets of strings (also called cross-bifix-free sets) have had an increasing interest in the frame of the researches about Theory of Codes. Recently some non-overlapping sets of strings with variable length were introduced. Moreover, the notion of non-overlapping strings has been naturally extended to the two dimensi...

We define a set of binary matrices where any two of them can not be placed one on the other in a way such that the corresponding entries coincide. The rows of the matrices are obtained by means of Dyck words. The cardinality of the set of such matrices involves Catalan numbers.

We define a set of matrices over a finite alphabet where all possible overlaps between any two matrices are forbidden. The set is also enumerated by providing some recurrences counting particular classes of restricted words. Moreover, we analyze the asymptotic cardinality of the set according to the parameters related to the construction of the mat...

In this paper we present a method to pass from a given recurrence relation with constant coefficients (in short, a C-finite recurrence) to a finite succession rule defining the same number sequence. Our method consists in two steps: first, we transform the given recurrence relation into an extended succession rule, then we provide a series of opera...

Two matrices are said non-overlapping if one of them can not be put on the other one in a way such that the corresponding entries coincide. We provide a set of non-overlapping binary matrices and a formula to enumerate it which involves the $k$-generalized Fibonacci numbers. Moreover, the generating function for the enumerating sequence is easily s...

Two matrices are said non-overlapping if one of them can not be put on the other one in a way such that the corresponding entries coincide. We provide a set of non-overlapping binary matrices and a formula to enumerate it which involves the $k$-generalized Fibonacci numbers. Moreover, the generating function for the enumerating sequence is easily s...

We study the construction and the enumeration of bit strings, or binary words in {0, 1}*, having more 1’s than 0’s and avoiding a set of Grand Dyck patterns which form a cross-bifix-free set. We give a particular jumping and marked succession rule which describes the growth of such words according to the number of 1’s. Then, we give the enumeration...

In this paper we present a method which can be used to investigate on the positivity of a number sequence defined by a recurrence relation having constant coefficients (in short, a \(C\)-recurrence).

A bidimensional bifix (in short bibifix) of a square matrix T is a square submatrix of T which occurs in the top-left and bottom-right corners of T. This allows us to extend the definition of bifix-free words and cross-bifix-free set of words to bidimensional structures. In this paper we exhaustively generate all the bibifix-free square matrices an...

A set of words with the property that no prefix of any word is the suffix of any other word is called cross-bifix-free set. We provide an efficient generating algorithm producing Gray codes for a remarkable family of cross-bifix-free sets.

Cross-bifix-free sets are sets of words such that no prefix of any word is a sufix of any other word. In this paper, we introduce a general constructive method for the sets of cross-bifix-free q-ary words of fixed length. It enables us to determine a cross-bifix-free words subset which has the property to be non-expandable.

Based on BRGC inspired order relations we give Gray codes and a generating algorithm for $q$-ary words avoiding a prescribed factor. These generalize an early 2001 result and a very recent one published by some of the present authors, and can be seen as an alternative to those of Squire published in 1996. Among the involved tools, we make use of ge...

A cross-bifix-free set of words is a set in which no prefix of any length of any word is the suffix of any other word in the set. A construction of cross-bifix-free sets has recently been proposed by Chee {\it et al.} in 2013 within a constant factor of optimality. We propose a \emph{trace partitioned} Gray code for these cross-bifix-free sets and...

We provide a trace partitioned Gray code for the set of q-ary strings avoiding a pattern constituted by k consecutive equal symbols. The definition of this Gray code is based on two different constructions, according to the parity of q. This result generalizes, and is based on, a Gray code for binary strings avoiding k consecutive 0's.

In this paper we consider the class of interval orders, recently considered by several authors from both an algebraic and an enumerative point of view. According to Fishburn’s Theorem (Fishburn J Math Psychol 7:144–149, 1970), these objects can be characterized as posets avoiding the poset 2 + 2. We provide a recursive method for the unique generat...

In this paper we study the construction and the enumeration of binary words in $\{0,1\}^*$ having more 1’s than 0’s and avoiding a set of cross-bifix-free patterns. 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. Moreover, the problem of a...

We introduce the notion of pattern in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all Dyck paths, which we call the Dyck pattern poset. Given a Dyck path P, we determine a formula for the number of...

A polyomino is said to be LL-convex if any two of its cells can be connected by a path entirely contained in the polyomino, and having at most one change of direction. In this paper, answering a problem posed by Castiglione and Vaglica [6], we prove that the class of LL-convex polyominoes is tiling recognizable. To reach this goal, first we express...

We introduce the notion of pattern in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all Dyck paths, which we call the Dyck pattern poset. Given a Dyck path P, we determine a formula for the number of...

In this paper we present a method to pass from a recurrence relation having constant coefficients (in short, a C-recurrence) to a finite succession rule defining the same number sequence. We recall that succession rules are a recently studied tool for the enumeration of combinatorial objects related to the ECO method. We also discuss the applicabil...

International audience We introduce the notion of $\textit{pattern}$ in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all Dyck paths, which we call the $\textit{Dyck pattern poset}$. Given a Dyck pat...

In this paper we propose an algorithm to generate binary words with no more 0's than 1's having a fixed number of 1's and avoiding the pattern $(10)^j1$ for any fixed $j \geq 1$. We will prove that this generation is exhaustive, that is, all such binary words are generated.

A permutominide is a set of cells in the plane satisfying special connectivity constraints and uniquely defined by a pair of permutations. It naturally generalizes the concept of permutomino, recently investigated by several authors and from different points of view [1, 2, 4, 6, 7]. In this paper, using bijective methods, we determine the enumerati...

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...

Cross-bifix-free sets are sets of words such that no prefix of any word is a suffix of any other word. In this paper, we introduce a general constructive method for the sets of cross-bifix-free binary words of fixed length. It enables us to determine a cross-bifix-free words subset which has the property to be non-expandable.

In this paper we study the enumeration and the construction, according to the number of ones, of particular binary words avoiding a fixed pattern. The growth of such words can be described by particular jumping and marked succession rules. This approach enables us to obtain an algorithm which constructs all binary words having a fixed number of one...

In [FP] the ECO method and Aigner's theory of Catalan-like numbers are compared, showing that it is often possible to trans-late a combinatorial situation from one theory into the other by means of a standard change of basis in a suitable vector space. In the present work we emphasize the soundness of such an approach by finding some applications s...

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...

A Catalan pair is a pair of binary relations (S,R) satisfying certain axioms. These objects are enumerated by the well-known Catalan numbers, and have been introduced with the aim of giving a common language to most of the structures counted by Catalan numbers. Here, we give a simple method to pass from the recursive definition of a generic Catalan...

We define the notion of a Catalan pair (which is a pair of binary relations (S,R) satisfying certain axioms) with the aim of giving a common language to several combinatorial interpretations of Catalan numbers. We show, in particular, that the second component R uniquely determines the pair, and we give a characterization of R in terms of forbidden...

Using the notion of series parallel interval order, we propose a unified setting to describe Dyck lattices and Tamari lattices (two well known lattice structures on Catalan objects) in terms of basic notions of the theory of posets. As a consequence of our approach, we find an extremely simple proof of the fact that the Dyck order is a refinement o...

We begin a systematic study of the enumerative combinatorics of mixed succession rules, i.e. succession rules such that, in the associated generating tree, nodes are allowed to produce sons at several different levels according to different production rules. Here we deal with a specific case, namely that of two different production rules whose rule...

We define the notion of a Catalan pair, which is a pair of (strict) order relations (S,R) satisfying certain axioms. We show that Catalan pairs of size n are counted by Catalan numbers. We study some combinatorial properties of the relations R and S. In particular, we show that the second component R uniquely determines the pair, and we give a char...

We define the notion of a Catalan pair (which is a pair of binary relations (S,R) satisfying certain axioms) with the aim of giving a common language to most of the combinatorial interpretations of Catalan numbers. We show, in particular, that the second component R uniquely determines the pair, and we give a characterization of R in terms of forbi...

In this paper we establish six bijections between a particular class of polyominoes, called deco polyominoes, enumerated according to their directed height by n!, and permutations. Each of these bijections allows us to establish different correspondences between classical statistics on deco polyominoes and on permutations.

We begin a systematic study of the enumerative combinatorics of mixed succession rules, which are succession rules such that, in the associated generating tree, the nodes are allowed to produce their sons at several different levels according to different production rules. Here we deal with a specific case, namely that of two different production r...

A permutomino of size n is a polynomino determined by particular pairs (π 1 ,π 2 ) of permutations of n, such that π 1 (i)≠π 2 (i), for 1≤i≤n. Here we study various classes of convex permutominoes. We determine some combinatorial properties and, in particular, the characterization for the permutations defining convex, directed-convex, and parallelo...

The method we have applied in "A. Bernini, L. Ferrari, R. Pinzani, Enumerating permutations avoiding three Babson-Steingrimsson patterns, Ann. Comb. 9 (2005), 137--162" to count pattern avoiding permutations is adapted to words. As an application, we enumerate several classes of words simultaneously avoiding two generalized patterns of length 3.

A permutomino of size n is a polyomino determined by particular pairs (P1, P2) of permutations of size n, such that P1(i) is different from P2(i), for all i. Here we determine the combinatorial properties and, in particular, the characterization for the permutations defining convex permutominoes. Using such a characterization, these permutations ca...

Starting from a succession rule for Catalan numbers, we define a procedure for encoding and listing the objects enumerated by these numbers such that two consecutive codes of the list differ only by one digit. The Gray code we obtain can be generalized to all the succession rules with the stability property: each label (k) has in its productions tw...

In this paper we determine a closed formula for the number of convex permutominoes of size n. We reach this goal by providing a recursive generation of all convex permutominoes of size n+1 from the objects of size n, according to the ECO method, and then translating this construction into a system of functional equations satisfied by the generating...

A permutomino of size n is a polyomino whose vertices define a pair of distinct permutations of length n. In this paper we treat various classes of convex permutominoes, including the parallelogram, the directed convex and the stack ones. Using bijective techniques we provide enumeration for each of these classes according to the size, and characte...

In this paper, we study the problem of determining digital sets by means of their X-rays. An X-ray of a digital set F in a direction u counts the number of points in F on each line parallel to u. A class 's of digital sets is characterized by the set U of directions if among all 's elements, each element in is determined by its X-rays in U's direct...

When planning a database, the problem of index selection is of particular interest. In this paper, we examine a transaction model which includes queries, updates, insertions and deletions, and we define a function that calculates the transactions total cost when an index set is used. Our aim was to minimize the function cost in order to identify th...

International audience In this paper we consider the class of $\textit{permutominoes}$, i.e. a special class of polyominoes which are determined by a pair of permutations having the same size. We give a characterization of the permutations associated with convex permutominoes, and then we enumerate various classes of convex permutominoes, including...

We consider posets of lattice paths (endowed with a natural order) and begin the study of such structures. We give an algebraic condition to recognize which ones of these posets are lattices. Next we study the class of Dyck lattices (i.e., lattices of Dyck paths) and give a recursive construction for them. The last section is devoted to the present...

The ECO method and the theory of Catalan-like numbers introduced by Aigner seems two completely unrelated combinatorial settings. In this work we try to establish a bridge between them, aiming at starting a (hopefully) fruitful study on their interactions. We show that, in a linear algebra context (more precisely, using infinite matrices), a succes...

We settle some conjectures formulated by A. Claesson and T. Mansour concerning generalized pattern avoidance of permutations. In particular, we solve the problem of the enumeration of permutations avoiding three generalized patterns of type (1, 2) or (2, 1) by using ECO method and a graphical representation of permutations.

Among the general methods usually employed in combinatorics (such as generating functions, theory of species, linear operator methods, order theory, incidence algebras, Hopf algebras, umbral calculus, and so on), I will deal with a particular one, which was born in Florence in the last decade of the past century, thanks to the work of a group of re...

The sum of the areas of the parallelogram polyominoes having semi-perimeter n+2 is equal to 4n. In this paper we give a simple proof of this property by means of a mapping from the cells of parallelogram polyominoes having semi-perimeter n+2 to the 4n words of length n of the free monoid {a,b,c,d}∗. This mapping works in linear time. Then, we intro...

We consider posets of lattice paths (endowed with a natural order) and begin the study of such structures. We give an algebraic condition to recognize which ones of these posets are lattices. Next we study the class of Dyck lattices (i.e., lattices of Dyck paths) and give a recursive construction for them. The last section is devoted to the present...

In this paper we apply the ECO method to the study of some enumerative properties of integer partitions. In so doing we both give an original description of some known constructions regarding partitions andpropose some results especially in the context of generalized hook partitions (i.e. partitions whose Ferrers diagrams fit inside of suitable hoo...

We study a generalization of the concept of succession rule, called jumping succession rule, where each label is allowed to produce its sons at different levels, according to the production of a fixed succession rule. By means of suitable linear algebraic methods, we obtain simple closed forms for the numerical sequences determined by such rules an...

Succession rules having a rational generating function are usually called rational succession rules. In this note we discuss some problems concerning rational succession rules, and determine a simple method to pass from a rational generating function to a rational succession rule, both de ning the same number sequence. 1

In the coordinate plane consider those lattice paths whose step types consist of (1,1), (1,−1), and perhaps one or more horizontal steps. For the set of such paths running from (0,0) to (n+2,0) and remaining strictly elevated above the horizontal axis elsewhere, we define a zeroth moment (cardinality), a first moment (essentially, the total area),...

In this paper we will give a formal description of succession rules in terms of linear operators satisfying certain conditions. This representation allows us to introduce a system of well-defined operations into the set of succession rules and then to tackle problems of combinatorial enumeration simply by using operators instead of generating funct...

We give an algebraic version of the enumeration of combinatorial objects (ECO) method, and of succession rules in general, by means of linear operators. Then, using the new algebraic notations, we translate some known results about the relationship between ECO-systems and generating functions into our language. Finally, we deal with the problem of...

For fixed positive integer k, let En denote the set of lattice paths using the steps (1, 1), (1, − 1), and (k, 0) and running from (0, 0) to (n, 0) while remaining strictly above the x-axis elsewhere. We first prove bijectively that the total area of the regions bounded by the paths of En and the x-axis satisfies a four-term recurrence depending on...

We present a family of number sequences which interpolates between the sequences Bn , of Bell numbers, and n!. It is defined in terms of permutations with forbidden patterns or subsequences. The introduction, as a parameter, of the number m of right-to-left minima yields an interpolation between Stirling numbers of the second kind S(n, m) and of th...

In this paper, we use ECO method and the concept of succession rule to enumerate restricted classes of combinatorial objects. Let Ω be the succession rule describing a construction of a combinatorial objects class, then the construction of the restricted class is described by means of an approximating succession ruleΩk obtained from Ω in a natural...

First, we explicit an infinite family of equivalent succession rules parametrized by a positive integer α, for which two specializations lead to the equivalence of rules defining the Catalan and Schröder numbers. Then, from an ECO-system for Dyck paths, we easily derive an ECO-system for complete binary trees, by using a widely known bijection betw...

In this paper, we study the problem of determining discrete sets by means of their X-rays. An X-ray of a discrete set F in a direction u counts the number of points in F on each line parallel to u. A class of discrete sets is characterized by the set U of directions if each element in is determined by its X-rays in the directions of U. By using the...

Vexillary permutations are very important for Schubert Polynomials. In this paper, we consider the enumeration of vexillary involutions, that is, 2143-avoiding involutions. Instead of solving the generating function obtained by a succession system characterizing vexillary involu-tions, we establish a one-to-one correspondence with 1-2 trees enumera...

A permutation avoids the subpattern ii has no subsequence having all the same pairwise comparisons as , and we write ∈ S(). We examine the classes of permuta-tions, S(321); S(321; 3 142) and S(4231; 4132), enumerated, respectively by the famous Catalan, Motzkin and Schr oder number sequences. We determine their generating functions according to the...

Un chemin de Schröder est un chemin positif du plan Z2 commençant et terminant sur l'axe des abscisses en effectuant des pas Nord-Est, Sud-Est, ou deux pas Est consécutifs. Nous donnons dans cet article un algorithme de complexité moyenne O(n) en espace et en temps qui engendre de façon aléatoire et uniforme un chemin de Schröder de longueur 2n. Si...

This paper deals with a study of the class of lattice paths, made of north, east, south, and west unit steps, which being at (−1, 0) and end at (0, 0), avoiding the non-negative x-axis. is bijectively proved to be enumerated by odd index Catalan numbers according to the number of steps.

In this paper we consider two classes of lattice paths on the plane which use north, east, south, and west unitary steps, beginning and ending at 0 0 . We enumerate them according to the number of steps by means of bijective arguments; in particular, we apply the cycle lemma. Then, using these results, we provide a bijective proof for the number of...

In this paper we study the class of generalized Motzkin paths with no hills and prove some of their combinatorial properties in a bijective way; as a particular case we have the Fine numbers, enumerating Dyck paths with no hills. Using the ECO method, we dene a recursive construction for Dyck paths such that the number of local expansions performed...

International audience In this paper we consider two classes of lattice paths on the plane which use \textitnorth, \textiteast, \textitsouth,and \textitwest unitary steps, beginningand ending at (0,0).We enumerate them according to the number ofsteps by means of bijective arguments; in particular, we apply the cycle lemma.Then, using these results,...

International audience A permutation π is said to be τ -avoiding if it does not contain any subsequence having all the same pairwise comparisons as τ . This paper concerns the characterization and enumeration of permutations which avoid a set F^j of subsequences increasing both in number and in length at the same time. Let F^j be the set of subsequ...

For every integer j>1, we dene a class of permutations in terms of certain forbidden sub- sequences. For j=1, the corresponding permutations are counted by the Motzkin numbers, and for j= 1 (dened in the text), they are counted by the Catalan numbers. Each value of j>1 gives rise to a counting sequence that lies between the Motzkin and the Catalan...

Basing on the ECO method, we define a recursive construction for Dyck and Schröder paths, respectively, such that the number of local expansions performed on each path depends on the number of its hills.

We introduce a system of well-defined operations on the set of succession rules. These operations allow us to tackle combinatorial enumeration problems simply by using succession rules instead of generating functions. Finally, we suggest several open problems the solution of which should lead to an algebraic characterization of the set of successio...

In this paper we use ECO method to enumerate restricted classes of combinatorial objects; if a succession rule Ω describes the construction of a class, then the restricted class can be described by means of an approximating succession rule Ω k obtained from Ω; we determine finite approximating rules for various classes of paths, and the approximati...

An elevated Schroder path is a lattice path that uses the steps (1; 1), (1; 1), and (2; 0), that begins and ends on the x-axis, and that remains strictly above the x-axis otherwise. The total area of elevated Schroder paths of length 2n + 2 satises the recurrence fn+1 =6 f n f n 1 , n2, with the initial conditions f0 =1 , f 1 =7 . A combinatorial i...

In this paper, we illustrate a method to enumerate two-dimensional directed animals, with compact sources, on both the square and the triangular lattice. We give a recursive description of these structures from which we deduce their generating function, according to various parameters: the area, the right half-width and the number of compact source...