Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
4.0
2.3
Journals
Mathematics
Special Issues
Combinatorics, Riordan Matrices and Umbral Calculus—in Memory of Prof. Emanuele...
share announcement

Combinatorics, Riordan Matrices and Umbral Calculus—in Memory of Prof. Emanuele Munarini

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "A: Algebra and Logic".

Deadline for manuscript submissions: 10 May 2025 | Viewed by 3387

Special Issue Editor

Prof. Dr. Donatella Merlini

E-Mail Website
Guest Editor
Dipartimento di Statistica, Informatica, Applicazioni (DiSIA), Università di Firenze, I-50134 Firenze, Italy
Interests: analysis of algorithms and data structures; enumerative combinatorics; symbolic computation; databases and data mining; educational data mining
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Combinatorial analysis is one of the branches of modern mathematics that is growing more and more rapidly. It has countless applications, ranging from enumerative and algebraic combinatorics, combinatorial number theory and graph theory to discrete mathematics, physics, computer science and biology.

Among the fundamental tools of combinatorial analysis are generating series, matrices and polynomial sequences. For instance, there is a deep connection between finite structures and generating functions, as witnessed by the theory of combinatorial species, as well as a strong interaction between enumeration, infinite matrices, formal series and sequences of polynomials, as seen in the theory of Riordan matrices and in umbral calculus.

This Special Issue accepts high-quality original contributions and review papers concerning theoretical aspects or applications relative to enumerative combinatorics, combinatorial matrices and polynomial sequences. Some of the main topics are listed below:

  1. Enumerative combinatorics.
  2. Generating functions.
  3. Combinatorial matrices, determinants and permanents.
  4. Hankel matrices and determinants.
  5. Riordan arrays and exponential Riordan arrays.
  6. Riordan group and its subgroups.
  7. Combinatorial number theory.
  8. Combinatorial identities.
  9. Umbral calculus.
  10. Sheffer sequences.
  11. Special functions and special polynomials.
  12. Hypergeometric series.
  13. Orthogonal polynomials.
  14. Q-calculus.
  15. Matrices and polynomials in graph theory.

I look forward to receiving your contributions.

Prof. Dr. Donatella Merlini
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • enumerative combinatorics
  • generating functions
  • combinatorial matrices
  • Riordan arrays
  • umbral calculus
  • Sheffer sequences

Benefits of Publishing in a Special Issue

  • Ease of navigation: Grouping papers by topic helps scholars navigate broad scope journals more efficiently.
  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
  • Expansion of research network: Special Issues facilitate connections among authors, fostering scientific collaborations.
  • External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
  • e-Book format: Special Issues with more than 10 articles can be published as dedicated e-books, ensuring wide and rapid dissemination.

Further information on MDPI's Special Issue policies can be found here.

Published Papers (5 papers)

Download All Papers
Order results
Result details
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:

Research

19 pages, 1730 KiB  
Article
Constrained Underdiagonal Paths and Pattern Avoiding Permutations
by Andrea Frosini, Veronica Guerrini and Simone Rinaldi
Mathematics 2025, 13(3), 517; https://doi.org/10.3390/math13030517 - 4 Feb 2025
Viewed by 376
Abstract
Moving from a simple bijection P between permutations Sn of length n and underdiagonal paths of size n, The we study and enumerate families of underdiagonal paths which are defined by restricting the bijection P to subclasses of Sn avoiding [...] Read more.
Moving from a simple bijection P between permutations Sn of length n and underdiagonal paths of size n, The we study and enumerate families of underdiagonal paths which are defined by restricting the bijection P to subclasses of Sn avoiding some vincular patterns. In particular, we will consider patterns of lengths 3 and 4, and, when it is possible, we will provide a characterization of the underdiagonal paths related to them in terms of geometrical constraints, or equivalently, the avoidance of some factors. Finally, we will provide a recursive growth of these families by means of generating trees and then their enumerative sequence. Full article
(This article belongs to the Special Issue Combinatorics, Riordan Matrices and Umbral Calculus—in Memory of Prof. Emanuele Munarini)
►▼ Show Figures

Figure 1

25 pages, 289 KiB  
Article
Extensions of Riordan Arrays and Their Applications
by Paul Barry
Mathematics 2025, 13(2), 242; https://doi.org/10.3390/math13020242 - 13 Jan 2025
Viewed by 517
Abstract
The Riordan group of Riordan arrays was first described in 1991, and since then, it has provided useful tools for the study of areas such as combinatorial identities, polynomial sequences (including families of orthogonal polynomials), lattice path enumeration, and linear recurrences. Useful extensions [...] Read more.
The Riordan group of Riordan arrays was first described in 1991, and since then, it has provided useful tools for the study of areas such as combinatorial identities, polynomial sequences (including families of orthogonal polynomials), lattice path enumeration, and linear recurrences. Useful extensions of the idea of a Riordan array have included almost Riordan arrays, double Riordan arrays, and their generalizations. After giving a brief overview of the Riordan group, we define two further extensions of the notion of Riordan arrays, and we give a number of applications for these extensions. The relevance of these applications indicates that these new extensions are worthy of study. The first extension is that of the reverse symmetrization of a Riordan array, for which we give two applications. The first application of this symmetrization is to the study of a family of Riordan arrays whose symmetrizations lead to the famous Robbins numbers as well as to numbers associated with the 20 vertex model of mathematical physics. We provide closed-form expressions for the elements of these arrays, and we also give a canonical Catalan factorization for them. We also describe an alternative family of Riordan arrays whose symmetrizations lead to the same integer sequences. The second application of this symmetrization process is to the area of the enumeration of lattice paths. We remain with the applications to lattice paths for the second extension of Riordan arrays that we introduce, which is the interleaved Riordan array. The methods used include generating functions, linear algebra, weighted compositions, and linear recurrences. In the case of the symmetrization process applied to Riordan arrays, we focus on the principal minor sequences of the resulting square matrices in the context of integrable lattice models. Full article
(This article belongs to the Special Issue Combinatorics, Riordan Matrices and Umbral Calculus—in Memory of Prof. Emanuele Munarini)
17 pages, 564 KiB  
Article
Clade Size Statistics Under Ford’s α-Model
by Antonio Di Nunzio and Filippo Disanto
Mathematics 2024, 12(24), 3974; https://doi.org/10.3390/math12243974 - 18 Dec 2024
Viewed by 492
Abstract
Given a labeled tree topology t of n taxa, consider a population P of k leaves chosen among those of t. The clade of P is the minimal subtree P^ of t containing P, and its size [...] Read more.
Given a labeled tree topology t of n taxa, consider a population P of k leaves chosen among those of t. The clade of P is the minimal subtree P^ of t containing P, and its size |P^| is provided by the number of leaves in the clade. We study distributive properties of the clade size variable |P^| considered over labeled topologies of size n generated at random in the framework of Ford’s α-model. Under this model, starting from the one-taxon labeled topology, a random labeled topology is produced iteratively by a sequence of α-insertions, each of which adds a pendant edge to either a pendant or internal edge of a labeled topology, with a probability that depends on the parameter α∈[0,1]. Different values of α determine different probability distributions over the set of labeled topologies of given size n, with the special cases α=0 and α=1/2 respectively corresponding to the Yule and uniform distributions. In the first part of the manuscript, we consider a labeled topology t of size n generated by a sequence of random α-insertions starting from a fixed labeled topology t∗ of given size k, and determine the probability mass function, mean, and variance of the clade size |P^| in t when P is chosen as the set of leaves of t inherited from t∗. In the second part of the paper, we calculate the probability that a set P of k leaves chosen at random in a Ford-distributed labeled topology of size n is monophyletic, that is, the probability that |P^|=k. Our investigations extend previous results on clade size statistics obtained for Yule and uniformly distributed labeled topologies. Full article
(This article belongs to the Special Issue Combinatorics, Riordan Matrices and Umbral Calculus—in Memory of Prof. Emanuele Munarini)
►▼ Show Figures

Figure 1

12 pages, 257 KiB  
Article
C-Finite Sequences and Riordan Arrays
by Donatella Merlini
Mathematics 2024, 12(23), 3671; https://doi.org/10.3390/math12233671 - 23 Nov 2024
Viewed by 440
Abstract
Many prominent combinatorial sequences, such as the Fibonacci, Lucas, Pell, Jacobsthal and Tribonacci sequences, are defined by homogeneous linear recurrence relations with constant coefficients. These sequences are often referred to as C-finite sequences, and a variety of representations have been employed throughout [...] Read more.
Many prominent combinatorial sequences, such as the Fibonacci, Lucas, Pell, Jacobsthal and Tribonacci sequences, are defined by homogeneous linear recurrence relations with constant coefficients. These sequences are often referred to as C-finite sequences, and a variety of representations have been employed throughout the literature, largely influenced by the author’s background and the specific application under consideration. Beyond the representation through recurrence relations, other approaches include those based on generating functions, explicit formulas, matrix exponentiation, the method of undetermined coefficients and several others. Among these, the generating function approach is particularly prevalent in enumerative combinatorics due to its versatility and widespread use. The primary objective of this work is to introduce an alternative representation grounded in the theory of Riordan arrays. This representation provides a general formula expressed in terms of the vectors of constants and initial conditions associated with any recurrence relation of a given order, offering a new perspective on the structure of such sequences. Full article
(This article belongs to the Special Issue Combinatorics, Riordan Matrices and Umbral Calculus—in Memory of Prof. Emanuele Munarini)
15 pages, 288 KiB  
Article
Multiple Sums of Circular Binomial Products
by Marta Na Chen and Wenchang Chu
Mathematics 2024, 12(12), 1855; https://doi.org/10.3390/math12121855 - 14 Jun 2024
Cited by 1 | Viewed by 700
Abstract
Five classes of multiple sums about circular products of binomial coefficients are investigated. Their generating functions are explicitly expressed as rational functions, with coefficients being Fibonacci and Lucas numbers. This is fulfilled by a novel approach called “recursive construction”, which also permits us [...] Read more.
Five classes of multiple sums about circular products of binomial coefficients are investigated. Their generating functions are explicitly expressed as rational functions, with coefficients being Fibonacci and Lucas numbers. This is fulfilled by a novel approach called “recursive construction”, which also permits us to establish, for the circular sums, both generating functions and recurrence relations through the Lagrange expansion formula. Full article
(This article belongs to the Special Issue Combinatorics, Riordan Matrices and Umbral Calculus—in Memory of Prof. Emanuele Munarini)
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying articles 1-5
Back to TopTop