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Symmetry
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Symmetry in Combinatorics and Discrete Mathematics
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Symmetry in Combinatorics and Discrete Mathematics

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 30 April 2025 | Viewed by 6073

Special Issue Editor

Prof. Dr. Takao Komatsu

E-Mail Website
Guest Editor
Faculty of Education, Nagasaki University, Nagasaki 852-8521, Japan
Interests: combinatorics; discrete mathematics; number theory; pure mathematics; special functions
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Many interesting symmetric properties appear in combinatorial numbers, including binomial coefficients, Stirling numbers and Bernoulli numbers. Such symmetric identities can be interpreted and studied from combinatorial, arithmetical, geometrical or analytical aspects. This Special Issue will present articles exploring new interpretations, applications and symmetrical and asymmetrical relations in combinatorics and discrete mathematics.

Prof. Dr. Takao Komatsu
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. Symmetry is an international peer-reviewed open access monthly 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 2400 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

  • infinite or finite groups
  • semigroups
  • designs and configurations
  • enumerative combinatorics
  • determinants and permanents
  • combinatorial sequences

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Published Papers (5 papers)

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Research

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15 pages, 425 KiB  
Article
Sequence of Bounds for Spectral Radius and Energy of Digraph
by Jietong Zhao, Saira Hameed, Uzma Ahmad, Ayesha Tabassum and Leila Asgharsharghi
Symmetry 2024, 16(10), 1386; https://doi.org/10.3390/sym16101386 - 18 Oct 2024
Viewed by 830
Abstract
The graph spectra analyze the structure of the graph using eigenspectra. The spectral graph theory deals with the investigation of graphs in terms of the eigenspectrum. In this paper, the sequence of lower bounds for the spectral radius of digraph D having at [...] Read more.
The graph spectra analyze the structure of the graph using eigenspectra. The spectral graph theory deals with the investigation of graphs in terms of the eigenspectrum. In this paper, the sequence of lower bounds for the spectral radius of digraph D having at least one doubly adjacent vertex in terms of indegree is proposed. Particularly, it is exhibited that ρ(D)≥αj=∑p=1m(χj+1(p))2∑p=1m(χj(p))2, such that equality is attained iff D=G↔+ {DE∉ Cycle}, where each component of associated graph is a k-regular or (k1,k2) semiregular bipartite. By utilizing the sequence of lower bounds of the spectral radius of D, the sequence of upper bounds of energy of D, where the sequence decreases when eU≤αj and increases when eU>αj, are also proposed. All of the obtained inequalities are elaborated using examples. We also discuss the monotonicity of these sequences. Full article
(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics)
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17 pages, 284 KiB  
Article
A Note on Incomplete Fibonacci–Lucas Relations
by Jingyang Zhong, Jialing Yao and Chan-Liang Chung
Symmetry 2023, 15(12), 2113; https://doi.org/10.3390/sym15122113 - 24 Nov 2023
Cited by 1 | Viewed by 931
Abstract
We define the incomplete generalized bivariate Fibonacci p-polynomials and the incomplete generalized bivariate Lucas p-polynomials. We study their recursive relations and derive an interesting relationship through their generating functions. Subsequently, we prove an incomplete version of the well-known Fibonacci–Lucas relation and [...] Read more.
We define the incomplete generalized bivariate Fibonacci p-polynomials and the incomplete generalized bivariate Lucas p-polynomials. We study their recursive relations and derive an interesting relationship through their generating functions. Subsequently, we prove an incomplete version of the well-known Fibonacci–Lucas relation and make some extensions to the relation involving incomplete generalized bivariate Fibonacci and Lucas p-polynomials. An argument about going from the regular to the incomplete Fibonacci–Lucas relation is discussed. We provide a relation involving the incomplete Leonardo and the incomplete Lucas–Leonardo p-numbers as an illustration. Full article
(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics)
11 pages, 299 KiB  
Article
More Variations on Shuffle Squares
by Jarosław Grytczuk, Bartłomiej Pawlik and Mariusz Pleszczyński
Symmetry 2023, 15(11), 1982; https://doi.org/10.3390/sym15111982 - 26 Oct 2023
Viewed by 1067
Abstract
We study an abstract variant of squares (and shuffle squares) defined by a constraint graph G, specifying which pairs of words form a square. So, a shuffle G-square is a word that can be split into two disjoint subwords U and W (of [...] Read more.
We study an abstract variant of squares (and shuffle squares) defined by a constraint graph G, specifying which pairs of words form a square. So, a shuffle G-square is a word that can be split into two disjoint subwords U and W (of the same length), which are joined by an edge. This setting generalizes a recently introduced model of shuffle squares based on word symmetry and permutations. By using the probabilistic method, we provide a sufficient condition for a constraint graph G guaranteeing the avoidability of shuffle G-squares. By a more-elementary method (known as Rosenfeld counting), we prove that G-squares are avoidable over an alphabet of size 4α, α>1, provided that the degree of every word of length n in G is at most αn. We also introduce the concept of the cutting distance between words and state several conjectures involving this notion and various kinds of shuffle squares. We suspect that, for every k⩾2, there is a constant ck such that every even word can be turned into a shuffle square by cutting it in at most ck places and rearranging the resulting pieces. We present some computational, as well as theoretical evidence in favor of this conjecture. Full article
(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics)
17 pages, 312 KiB  
Article
The p-Numerical Semigroup of the Triple of Arithmetic Progressions
by Takao Komatsu and Haotian Ying
Symmetry 2023, 15(7), 1328; https://doi.org/10.3390/sym15071328 - 29 Jun 2023
Cited by 3 | Viewed by 1035
Abstract
For given positive integers a1,a2,⋯,ak with gcd(a1,a2,⋯,ak)=1, the denumerant [...] Read more.
For given positive integers a1,a2,⋯,ak with gcd(a1,a2,⋯,ak)=1, the denumerant d(n)=d(n;a1,a2,⋯,ak) is the number of nonnegative solutions (x1,x2,⋯,xk) of the linear equation a1x1+a2x2+⋯+akxk=n for a positive integer n. For a given nonnegative integer p, let Sp=Sp(a1,a2,⋯,ak) be the set of all nonnegative integer n’s such that d(n)>p. In this paper, by introducing the p-numerical semigroup, where the set N0\Sp is finite, we give explicit formulas of the p-Frobenius number, which is the maximum of the set N0\Sp, and related values for the triple of arithmetic progressions. The main aim is to determine the elements of the p-Apéry set. Full article
(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics)

Review

Jump to: Research

29 pages, 396 KiB  
Review
Reciprocal Symmetry via Inverse Series Pairs
by Wenchang Chu
Symmetry 2023, 15(5), 1086; https://doi.org/10.3390/sym15051086 - 15 May 2023
Viewed by 1380
Abstract
Reciprocal series are employed to systematically review convolution sums, orthogonality relations, recurrence relations and reciprocal formulae for several classical number sequences, such as binomial coefficients, Stirling numbers, Bernoulli numbers, and Euler numbers. Full article
(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics)
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