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A Heffter array over an additive group $G$ is any partially filled array $A$ satisfying that: (1) each one of its rows and columns sum to zero in $G$, and (2) if $i\in G\setminus\{0\}$, then either $i$ or $-i$ appears exactly once in $A$. In this paper, this notion is naturally generalized to that of $\mathcal{B}$-Heffter array over a partial loop, where $\mathcal{B}$ is a set of block-sum polynomials over an affine $1$-design on the set of entries in $A$.

The Hadamard quasigroup product has recently been introduced as a natural generalization of the classical Hadamard product of matrices. It is defined as the superposition operator of three binary operations, one of them being a quasigroup operation. This paper delves into the fundamentals of this superposition operator by considering its more general version over multiary groupoids. Particularly, we show how this operator preserves algebraic identities, multiary groupoid structures, inverse elements, isotopes, conjugates and orthogonality. Then, we generalize the mentioned Hadamard quasigroup product to multiary quasigroups. Based on this product, we prove that the number of $m$-ary quasigroups defined on a given set $X$ coincides with the number of $m$-ary operations that are orthogonal to a given $m$-set of orthogonal $m$-ary operations over $X$.

Now-a-days, ensuring data security has become an increasingly formidable challenge in safeguarding individuals' sensitive information. Secret-sharing scheme has evolved as a most successful cryptographic technique that allows a secret to be divided or distributed among a group of participants in such a way that only a subset of those participants can reconstruct the original secret. This provides a safe level of security and redundancy, ensuring that no single individual possesses the complete secret. The implementation of Rainbow Antimagic coloring within these schemes not only safeguards the data but also ensures an advanced level of information security among multi-participant groups. Additionally, the retrieved data is reconstructed and can be disseminated to all group participants via multiple rounds of communication.

Let $G$ and $H$ be two graphs, each one of them being a path, a cycle or a star. In this paper, we determine the $b$-chromatic number of every subdivision-vertex neighbourhood corona $G\boxdot H$ or $G\boxdot K_n$, where $K_n$ is the complete graph of order $n$. It is also established for those graphs $K_n\boxdot G$ having $m$-degree not greater than $n+2$. All the proofs are accompanied by illustrative examples., Comment: 25 pages, 24 figures

In this paper, a new family of rotationally symmetric planar graphs is described based on an edge coalescence of planar chorded cycles. Their local fractional metric dimension is established for those ones arisen from chorded cycles of order up to six. Their asymptotic behaviour enables us to ensure the existence of new families of rotationally symmetric planar graphs with either constant or bounded local fractional dimension., Comment: 20 pages, 8 figures and 8 tables

Computing the autotopism group of a partial Latin rectangle can be performed in a variety of ways. This pilot study has two aims: (a) to compare these methods experimentally, and (b) to identify the design goals one should have in mind for developing practical software. To this end, we compare six families of algorithms (two backtracking methods and four graph automorphism methods), with and without the use of entry invariants, on two test suites. We consider two entry invariants: one determined by the frequencies of row, column, and symbol representatives, and one determined by $2 \times 2$ submatrices. We find: (a) with very few entries, many symmetries often exist, and these should be identified mathematically rather than computationally, (b) with an intermediate number of entries, a quick-to-compute entry invariant was effective at reducing the need for computation, (c) with an almost-full partial Latin rectangle, more sophisticated entry invariants are needed, and (d) the performance for (full) Latin squares is significantly poorer than other partial Latin rectangles of comparable size, obstructed by the existence of Latin squares with large (possibly transitive) autotopism groups., Comment: 16 pages, 5 figures, 1 table

This paper deals with distinct computational methods to enumerate the set $\mathrm{PLR}(r,s,n;m)$ of $r \times s$ partial Latin rectangles on $n$ symbols with $m$ non-empty cells. For fixed $r$, $s$, and $n$, we prove that the size of this set is a symmetric polynomial of degree $3m$, and we determine the leading terms (the monomials of degree $3m$ through $3m-9$) using inclusion-exclusion. For $m \leq 13$, exact formulas for these symmetric polynomials are determined using a chromatic polynomial method. Adapting Sade's method for enumerating Latin squares, we compute the exact size of $\mathrm{PLR}(r,s,n;m)$, for all $r \leq s \leq n \leq 7$, and all $r \leq s \leq 6$ when $n=8$. Using an algebraic geometry method together with Burnside's Lemma, we enumerate isomorphism, isotopism, and main classes when $r \leq s \leq n \leq 6$. Numerical results have been cross-checked where possible., Comment: 36 pages, 2 figures, 15 tables

Every partial colouring of a Hamming graph is uniquely related to a partial Latin hyper-rectangle. In this paper we introduce the $\Theta$-stabilized $(a,b)$-colouring game for Hamming graphs, a variant of the $(a,b)$-colouring game so that each move must respect a given autotopism $\Theta$ of the resulting partial Latin hyper-rectangle. We examine the complexity of this variant by means of its chromatic number. We focus in particular on the bi-dimensional case, for which the game is played on the Cartesian product of two complete graphs, and also on the hypercube case., Comment: 23 pages, 7 figures

The mitosis process of an eukaryotic cell can be represented by the structure constants of an evolution algebra. Any isotopism of the latter corresponds to a mutation of genotypes of the former. This paper uses Computational Algebraic Geometry to determine the distribution of three-dimensional evolution algebras over any field into isotopism classes and hence, to describe the spectrum of genetic patterns of three distinct genotypes during a mitosis process. Their distribution into isomorphism classes is also determined in case of dealing with algebras having a one-dimensional annihilator., Comment: 17 pages. . Accepted for publication in Applied Mathematics and Computation

This paper provides an in-depth analysis of how computational algebraic geometry can be used to deal with the problem of counting and classifying $r\times s$ partial Latin rectangles based on $n$ symbols of a given size, shape, type or structure. The computation of Hilbert functions and triangular systems of radical ideals enables us to solve this problem for all $r,s,n\leq 6$. As a by-product, explicit formulas are determined for the number of partial Latin rectangles of size up to six. We focus then on the study of non-compressible regular partial Latin squares and their equivalent incidence structure called seminet, whose distribution into main classes is explicitly determined for point rank up to eight. We prove in particular the existence of two new configurations of point rank eight., Comment: 28 pages, 6 figures, 7 tables

The current paper deals with the enumeration and classification of the set $\mathcal{SOR}_{r,n}$ of self-orthogonal $r\times r$ partial Latin rectangles based on $n$ symbols. These combinatorial objects are identified with the independent sets of a Hamming graph and with the zeros of a radical zero-dimensional ideal of polynomials, whose reduced Gr\"obner basis and Hilbert series can be computed to determine explicitly the set $\mathcal{SOR}_{r,n}$. In particular, the cardinality of this set is shown for $r\leq 4$ and $n\leq 9$ and several formulas on the cardinality of $\mathcal{SOR}_{r,n}$ are exposed, for $r\leq 3$. The distribution of $r\times s$ partial Latin rectangles based on $n$ symbols according to their size is also obtained, for all $r,s,n\leq 4$., Comment: 15 pages, 1 figure, 4 tables. Accepted for publication in European Journal of Combinatorics