# Louis W. ShapiroHoward University | HU · Department of Mathematics

Louis W. Shapiro

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77

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## Publications

Publications (77)

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-inv...

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...

Involutions and pseudo-involutions in the Riordan group are interesting because of their numerous applications. In this paper we study involutions using sequence characterizations of the Riordan arrays. For a given B-sequence we find the unique function f(z) such that the array(g(z), f(z)) is a pseudo-involution. As a combinatorial application, we...

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...

We consider ordered trees with a distinguished vertex which we call a mutator. There are many situations where this model arises. An ordered tree could represent a river network, supply lines, an employee organization chart, a phylogenetic tree, or a family tree. The mutator could be a dam or a source of pollution, a break in a supply line, a corru...

The r-ary number sequences given by (formula presented) are analogs of the sequence of the Catalan numbers 1 (formula presented). Their his- tory goes back at least to Lambert [8] in 1758 and they are of considerable interest in sequential testing. Usually, the sequences are considered sep- arately and the generalizations can go in several directio...

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...

In this paper we compute the distribution of several statistics on the set of rooted ordered trees. In particular, we determine the number of boundary edges, the number of singleton boundary edges, and the analogous values when edges may take on one of k colors.

A mutation will affect an individual and some or all of its descendants. In
this paper, we investigate ordered trees with a distinguished vertex called the
mutator. We describe various mutations in ordered trees, and find the
generating functions for statistics concerning trees with those mutations. The
examples give new interpretations to several...

Stirling numbers and Bessel numbers have a long history, and both have been generalized in a variety of directions. Here, we present a second level generalization that has both as special cases. This generalization often preserves the inverse relation between the first and second kind, and has simple combinatorial interpretations. We also frame the...

In theory, Riordan arrays can have any AA-sequence and any ZZ-sequence. For examples of combinatorial interest they tend to be related. Here we look at the case that they are identical or nearly so. We provide a combinatorial interpretation in terms of weighted Łukasiewicz paths and then look at several large classes of examples.

In this paper, we describe the uplift principle for ordered trees which lets us solve a variety of combinatorial problems in two simple steps. The first step is to find the appropriate generating function at the root of the tree, the second is to lift the result to an arbitrary vertex by multiplying by the leaf generating function. This paper, thou...

An ordered tree, also known as a plane tree or a planar tree, is defined recursively, as having a root and an ordered set of subtrees. An oldest child tree is an ordered tree where the rightmost edge above any vertex can either red or green. If we are considering family trees, then oldest child can be spoiled or not. In this paper we show that the...

The hitting time subgroup is a subgroup of the Riordan group and it came up recently in connection with Faber polynomials as studied in classical complex analysis. It turns out that the hitting time subgroup also arises in situations involving random walks and often is the distinguished leaf generating function for classes of ordered trees. These c...

The Riordan group is a group of infinite lower triangular matrices that are defined by two generating functions, g and f. The kth column of the matrix has the generating function gf k . In the Double Riordan group there are two generating function f 1 and f 2 such that the columns, starting at the left, have generating functions using f 1 and f 2 a...

The lattice polynomials $L_{i,j}(x)$ are introduced by Hough and Shapiro as a weighted count of certain lattice paths from the origin to the point $(i,j)$. In particular, $L_{2n, n}(x)$ reduces to the generating function of the numbers $T_{n,k}={1\over n}{n-1+k\choose n-1}{2n-k\choose n+1}$, which can be viewed as a refinement of the $3$-Catalan nu...

Let Σ be the set of functions, convergent for all |z|>1, with a Laurent series of the form f(z)=z+∑n⩾0anz-n. In this paper, we prove that the set of Faber polynomial sequences over Σ and the set of their normalized kth derivative sequences form groups which are isomorphic to the hitting time subgroup and the Bell(k) subgroup of the Riordan group, r...

We give a short combinatorial proof of a Fine number generating function identity and then explore some of the ramifications in terms of random walks, friendly walkers, and ordered trees. The results are also generalized to obtain similar results including those in Motzkin and Schröder settings.

In this paper, we study symmetric lattice paths. Let $d_{n}$, $m_{n}$, and $s_{n}$ denote the number of symmetric Dyck paths, symmetric Motzkin paths, and symmetric Schr\"oder paths of length $2n$, respectively. By using Riordan group methods we obtain six identities relating $d_{n}$, $m_{n}$, and $s_{n}$ and also give two of them combinatorial pro...

Several important combinatorial arrays, after inserting some minus signs, turn out to be involutions when considered as lower triangular matrices. Among these are the Pascal, RNA, and directed animal matrices. These examples and many others are in the Bell subgroup of the Riordan group. We characterize all such pseudo-involutions by means of a sing...

In this paper, we obtain a generalized Lucas polynomial sequence from the lattice paths for the Delannoy numbers by allowing weights on the steps (1,0),(0,1) and (1,1). These weighted lattice paths lead us to a combinatorial interpretation for such a Lucas polynomial sequence. The concept of Riordan arrays is extensively used throughout this paper.

A new bijection between 3-Motzkin paths and Schröder paths with no peak at odd height is presented, together with numerous consequences involving related combina-torial structures such as 2-Motzkin paths, ordinary Motzkin paths and Dyck paths.

In this note we start by computing the average number of protected points in all ordered trees with n edges. This can serve as a guide in various organizational schemes where it may be desirable to have a large or small number of protected points. We will also look a few subclasses with a view to increasing or decreasing the proportion of protected...

We study involutions in the Riordan group, especially those with combinatorial meaning. We give a new determinantal criterion for a matrix to be a Riordan involution and examine several classes of examples. A complete characterization of involutions in the Appell subgroup is developed. In another direction we find several examples that generalize t...

Predicting the secondary structure of an RNA sequence is an important problem in structural bioinformatics. The general RNA folding problem, where the sequence to be folded may contain pseudoknots, is computationally intractable when no prior knowledge on the pseudoknot structures the sequence contains is available. In this paper, we consider stabl...

A more suggestive title would be "Structures counted by certain Fibonacci-like sequences", as the sequence has the Fibonacci numbers as a special case when d = 1 with suitable initial conditions. We find a number of different objects enumerated by these sequences, including skinny trees, tilings with dominoes and lattice paths.

The Schröder relations are put into correspondence with a variety of other objects of combinatorial interest, including bushes,
foliated trees, lattice paths with diagonal steps and certain sorts of two-coloured objects. A distinction between left and
right Schröder relations is used in obtaining some of these correspondenced; others involve the us...

A double ended queue or deque is a linear list for which all insertions and deletions occur at the ends of the list. We give
a direct, ‘pictorial’ proof of a result of Knuth on the enumeration of permutations obtainable from output restricted deques.
This approach readily identifies the numbers of these permutations as Schröder numbers and leads na...

We introduce the notion of doubly rooted plane trees and give a decomposition of these trees, called the butterfly decomposition, which turns out to have many applications. From the butterfly decomposition we obtain a one-to-one correspondence between doubly rooted plane trees and free Dyck paths, which implies a simple derivation of a relation bet...

We give a combinatorial interpretation of a matrix identity on Catalan numbers and the sequence $(1, 4, 4^2, 4^3, ...)$ which has been derived by Shapiro, Woan and Getu by using Riordan arrays. By giving a bijection between weighted partial Motzkin paths with an elevation line and weighted free Motzkin paths, we find a matrix identity on the number...

The purpose of this paper is twofold. As the first goal, we show that three different classes of random walks are counted by the Pell numbers. The calculations are done using a convenient technique that involves the Riordan group. This leads to the second goal, which is to demonstrate this convenient technique. We also construct bijections among Pe...

The problem of counting plane trees with $n$ edges and an even or an odd number of leaves was studied by Eu, Liu and Yeh, in connection with an identity on coloring nets due to Stanley. This identity was also obtained by Bonin, Shapiro and Simion in their study of Schr\"oder paths, and it was recently derived by Coker using the Lagrange inversion f...

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...

Noncrossing trees are trees formed by arranging n points around a circle and using these points as the vertices of a tree whose edges do not cross. The number of such trees is the ternary number, T n =1 2n+13n n. We refine this count by considering number of descents and degree at the root. We first show that the numbers involved give an element of...

A new bijection between ordered trees and 2-Motzkin paths is presented, together with its numerous consequences regarding ordered trees as well as other combinatorial structures such as Dyck paths, bushes, {0,1,2}-trees, Schröder paths, RNA secondary structures, noncrossing partitions, Fine paths, and Davenport–Schinzel sequences.RésuméUne nouvelle...

The Fine numbers and the Catalan numbers are intimately related. Two manifestations are the identity C-n = 2F(n) + Fn-1, n greater than or equal to 1, and the generating function identities F = C/(1 + zC), C = F/ (1 - zF). In this paper we collect and organize the previous literature, present many new settings, and develop the theory and generating...

We want to consider some infinite arithmetic triangles and the shape of the entries in the nth row. More specifically, after dividing by the row sum, we show that the nth row approaches a normal distribution. The tools we will use are matching polynomials and Chebyshev polynomials. Here are the definitions and the few basic facts that we will need....

In the "tennis ball" problem we are given successive pairs of balls numbered (1,2), (3,4),... At each stage we throw one ball out of the window. After n stages some set of n balls is on the lawn. We find a generating function and a closed formula for the sequence 3, 23, 131, 664, 3166, 14545, 65187, 287060, 1247690,..., the n-th term of which gives...

For n 1, each of the Schroder numbers, s n = 1; 1; 3; 11; 45; : : : counts the possible generalized bracketings on a word of n letters. Each of the large Schroder numbers, r n = 1; 2; 6; 22; 90; : : : , counts the lattice paths running from (0; 0) to (n Gamma 1; n Gamma 1), using the steps (0; 1), (1; 1), and (1; 0), and never passing below the lin...

We present enumerative results concerning plane lattice paths starting at the origin, with steps (1,0), (1,1) and (0,1). Such paths with a specified endpoint are counted by the Delannoy numbers, while those paths which in addition do not run above the line y=x are counted by the Schröder numbers. We develop q-analogues of the Delannoy and Schröder...

In this note, a new method for taking the first few terms of a sequence and making an educated guess as to the generating function of the sequence is described. The method involves a matrix factorization into lower triangular, diagonal, and upper triangular matrices (the LDU decomposition), generating functions, and solving a first-order differenti...

A percolation process on n x n 0-1 matrices is defined. This process is defined so that a zero entry becomes one if two or more of its neighbors have the value one. Entries that have the value one never change. The process halts when no more entries can change. The initial matrices are taken to be all the n x n permutation matrices. It is shown tha...

In this note we present a combinatorial proof from first principles of the Hurwitz formula which is a generalization of Abel's formula which in turn is a deep generalization of the binomial formula.

The moments of a Catalan triangle are computed. The method is similar to that used for probability generating functions. This however does not explain the elegance of the results until further combinatorial connections are developed. These include Eulerian numbers, runs, zig-zag permutations, tangent numbers and a kind of nontrivial run called a sl...

The purpose of this note is to give a short proof of a result published by J. Rosen (J. Combin. Theory Ser. A20 (1976), 377–384) a few years ago relating ballot paths and the tangent numbers. A similar proof of a similar result with the secant numbers is also outlined.

A certain Markov chain which was encountered by T. L. Hill in the study of the kinetics of a linear array of enzymes is studied. An explicit formula for the steady state probabilities is given and some conjectures raised by T. L. Hill are proved.

Two equations relate the well-known Catalan numbers with the relatively unknown Motzkin numbers which suggest that the combinatorial settings of the Catalan numbers should also yield Motzkin numbers. In this paper we provide a representative selection of 14 situations where the Motzkin numbers occur along with the Catalan numbers.

We develop an arithmetic triangle similar to Pascal's triangle. The entries are interpreted in terms of numbers of pairs of nonintersecting paths in the first quadrant. The main applications are results about the Catalan numbers and various random walk problems.

In this paper a very short proof is given of an identity concerning Catalan numbers due originally to Touchard.

## Projects

Project (1)