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Paul Barry

Publications and source records attributed to Paul Barry.

At least 37 records · Page 2Linked to original sources
arXiv2021

Centered polygon numbers, heptagons and nonagons, and the Robbins numbers

In this note, we explore certain determinantal descriptions of the Robbins numbers. Techniques used for this include continued fractions, Riordan arrays and series inversion. Proven and conjectured representations involve the determinants of both Hankel and symmetric matrices. In specific cases, links are drawn to centered polygonal numbers, and to heptagons and nonagons. We conjecture a Hankel transform determinant for the Robbins numbers related to the Fibonacci and the Catalan numbers.

math.CO↗
arXiv2021

On the Gap-sum and Gap-product Sequences of Integer Sequences

In this note, we explore two families of sequences associated to a suitable integer sequence: the gap-sum sequence and the gap-product sequence. These are the sums and the products of consecutive numbers not in the original sequence. We give closed forms for both, in terms of the original sequence, and in the case of Horadam sequences, we find the generating function of the gap-sum sequence. For some elementary sequences, we indicate that the gap-product sequences are given by the Fuss-Catalan-Raney numbers.

math.CO↗
arXiv2021

Jointly Learning Clinical Entities and Relations with Contextual Language Models and Explicit Context

We hypothesize that explicit integration of contextual information into an Multi-task Learning framework would emphasize the significance of context for boosting performance in jointly learning Named Entity Recognition (NER) and Relation Extraction (RE). Our work proves this hypothesis by segmenting entities from their surrounding context and by building contextual representations using each independent segment. This relation representation allows for a joint NER/RE system that achieves near state-of-the-art (SOTA) performance on both NER and RE tasks while beating the SOTA RE system at end-to-end NER & RE with a 49.07 F1.

cs.CL↗
arXiv2021

On the inversion of Riordan arrays

Many Riordan arrays play a significant role in algebraic combinatorics. We explore the inversion of Riordan arrays in this context. We give a general construct for the inversion of a Riordan array, and study this in the case of various subgroups of the Riordan group. For instance, we show that the inversion of an ordinary Bell matrix is an exponential Riordan array in the associated subgroup. Examples from combinatorics and algebraic combinatorics illustrate the usefulness of such inversions. We end with a brief look at the inversion of exponential Riordan arrays. A final example places Airey's convergent factor in the context of a simple exponential Riordan array.

math.CO↗
arXiv2020

The second production matrix of a Riordan array

A Riordan array is defined by its production matrix. In this paper, we explore the notion of the second production matrix of a Riordan array, characterizing the matrix that is generated by it. We indicate how this procedure can be generalized to the so-called $n$-th production matrix of a Riordan array.

math.CO↗
arXiv2020

Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers

We provide a context around a conjectured closed form for the Hankel transform of linear combinations of consecutive pairs of Catalan numbers. This generalizes the formula for the Hankel transforms of the shifted Catalan numbers and the known results for linear combinations of pairs of Catalan numbers. Many interesting number triangles emerge, some of which we analyze using the language of Riordan arrays.

math.CO↗
arXiv2020

On the Central Description of the Group of Riordan Arrays

We provide an alternative description of the group of Riordan arrays, by using two power series of the form $\sum_{n=0}^{\infty} g_n x^n$, where $g_0 \ne 0$ to build a typical element of the constructed group. We relate these elements to Riordan arrays in the usual description, showing that each newly constructed element is the vertical half of a "usual" element. The product rules and the construction of the inverse are given in this new description, which we call a "central" description, because of links to the central coefficients of Riordan arrays. This is done for the case of ordinary generating functions. Finally, we briefly look at the exponential case.

math.CO↗
arXiv2020

Some observations on the Rueppel sequence and associated Hankel determinants

Starting with a definition based on the Catalan numbers, we carry out an empirical study of the Rueppel sequence. We use the Hankel transform as the main technique. By means of this transform we find links to such sequences as the Jacobi sequence and the paper-folding sequence. We also study several number arrays defined by this sequence, which are analogs of the important Catalan triangles of combinatorics. We identify a sequence related to the Golay-Rudin-Shapiro sequence that plays a fundamental role in linking the Hankel transforms of the paper with classical sequences. Examples are given where certain privileged permutations appearing in the Leibnitz formula for determinants have a special role.

math.CO↗
arXiv2020

On a Central Transform of Integer Sequences

We use the concept of the half of a lower-triangular matrix to define a transformation on integer sequences. We explore the properties of this transformation, including in some cases a study of the Hankel transform of the transformed sequences. Starting from simple sequences with elementary rational generating functions, we obtain many sequences of combinatorial significance. We make extensive use of techniques drawn from the theory of Riordan arrays.

math.CO↗
arXiv2020

Characterizations of the Borel triangle and Borel polynomials

We use Riordan array theory to give characterizations of the Borel triangle and its associated polynomial sequence. We show that the Borel polynomials are the moment sequence for a family of orthogonal polynomials whose coefficient array is a Riordan array. The role of the Catalan matrix in defining the Borel triangle is examined. We generalize the Borel triangle to a family of two parameter triangles. Generating functions are expressed as Jacobi continued fractions, as well as the zeros of appropriate quadratic expressions. The Borel triangle is exhibited as a Hadamard product of matrices. We investigate the reversions of the triangles studied. We introduce the notion of Fuss-Borel triangles and Fuss-Catalan triangles. We end with some remarks on the Catalan triangle.

math.CO↗
arXiv2019

Chebyshev moments and Riordan involutions

We show that the coefficient array of a family of Chebyshev moments defines an involution in the group of Riordan arrays. We then extend this result to certain families of $d$-orthogonal polynomials.

math.CO↗
arXiv2019

A Note on Riordan Arrays with Catalan Halves

For a lower triangular matrix $(t_{n,k})$ we call the matrices with respective entries $(t_{2n-k,n})$ and $(t_{2n,n+k})$ the vertical and the horizontal halves. In this note, we discuss Riordan arrays whose halves are closely related to the Catalan matrices.

math.CO↗
arXiv2019

Riordan arrays, the $A$-matrix, and Somos $4$ sequences

We characterize certain Riordan arrays by their $A$-matrices and $ρ$ sequences. We conjecture the form of a generic $A$-matrix which leads to Somos $4$ sequences. We find an $A$-matrix that produces a Riordan quasi-involution, and we study the $A$-matrices and $ρ$ sequences of the moment matrices of certain perturbed orthogonal polynomials.

math.CO↗
arXiv2019

Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials

We show that the Catalan-Schroeder convolution recurrences and their higher order generalizations can be solved using Riordan arrays and the Catalan numbers. We investigate the Hankel transforms of many of the recurrence solutions, and indicate that Somos $4$ sequences often arise. We exhibit relations between recurrences, Riordan arrays, elliptic curves and Somos $4$ sequences. We furthermore indicate how one can associate a family of orthogonal polynomials to a point on an elliptic curve, whose moments are related to recurrence solutions.

math.CO↗
arXiv2019

On the halves of a Riordan array and their antecedents

Every Riordan array has what we call a horizontal half and a vertical half. These halves of a Riordan array have been studied separately before. Here, we place them in a common context, showing that one may be obtained from the other. We also ask and answer the question: given a Riordan array, when is it the half (either horizontal of vertical) of a Riordan array?

math.CO↗
arXiv2019

Constant coefficient Laurent biorthogonal polynomials, Riordan arrays and moment sequences

We study properties of constant coefficient Laurent biorthogonal polynomials using Riordan arrays. We give details of related orthogonal polynomials, and we explore relationships between the moments of these orthogonal polynomials, the moments of the defining Laurent biorthogonal polynomials, and the expansions of $T$-fractions. Closed form expressions are given for the polynomials and their moments.

math.CO↗
arXiv2019

On the $r$-shifted central triangles of a Riordan array

Let $A$ be a proper Riordan array with general element $a_{n,k}$. We study the one parameter family of matrices whose general elements are given by $a_{2n+r, n+k+r}$. We show that each such matrix can be factored into a product of a Riordan array and the original Riordan array $A$, thus exhibiting each element of the family as a Riordan array. We find transition relations between the elements of the family, and examples are given. Lagrange inversion is used as a main tool in the proof of these results.

math.CO↗