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With this work we aim to show how Mathematica can be a useful tool to investigate properties of combinatorial structures. Specifically, we will face enumeration problems on independent subsets of powers of paths and cycles, trying to…
To study orthogonal arrays and signed orthogonal arrays, Ray-Chaudhuri and Singhi (1988 and 1994) considered some module spaces. Here, using a linear algebraic approach we define an inclusion matrix and find its rank. In the special case of…
In 2008, Cavenagh and Dr\'{a}pal, et al, described a method of constructing Latin trades using groups. The Latin trades that arise from this construction are entry-transitive (that is, there always exists an autoparatopism of the Latin…
A latin bitrade is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. Dr\'apal (\cite{Dr9}) showed that a latin bitrade is…
Latin squares are interesting combinatorial objects with many applications. When working with Latin squares, one is sometimes led to deal with partial Latin squares, a generalization of Latin squares. One of the problems regarding partial…
In our article we consider some algebraical methods which may be useful in some inverse spectral problems. The reconstraction of the matrix from its minors is considered.
By simulating an ergodic Markov chain whose stationary distribution is uniform over the space of nxn Latin squares, Mark T. Jacobson and Peter Matthews [4], have discussed elegant methods by which they generate Latin squares with a uniform…
We propose a construction of lattices from (skew-) polynomial codes, by endowing quotients of some ideals in both number fields and cyclic algebras with a suitable trace form. We give criteria for unimodularity. This yields integral and…
Many combinatorial problems can be formulated as a polynomial optimization problem that can be solved by state-of-the-art methods in real algebraic geometry. In this paper we explain many important methods from real algebraic geometry, we…
From the matrix point of view, we use the recursion to discuss four combinatorial numbers in terms of the integer lattice paths, this is different from Andr\'a's method (Andra). We give four tables and matrices, and their relations, and…
A Latin square of order $n$ is an $n\times n$ matrix in which each row and column contains each of $n$ symbols exactly once. For $\epsilon>0$, we show that with high probability a uniformly random Latin square of order $n$ has no proper…
This is an exercise based approach to matrix groups. The idea is to collect a bunch of exercises at one place which anyone with basic knowledge of linear algebra can attempt to solve and learn matrix groups and algebraic groups.
In this paper, we present a generalized version of the matrix chain algorithm to generate efficient code for linear algebra problems, a task for which human experts often invest days or even weeks of works. The standard matrix chain problem…
We explore the connections between the linear algebra of symmetric matrices over GF(2) and the circuit theory of 4-regular graphs. In particular, we show that the equivalence relation on simple graphs generated by local complementation can…
In this paper, the compact linearization approach originally proposed for binary quadratic programs with assignment constraints is generalized to such programs with arbitrary linear equations and inequalities that have positive coefficients…
We briefly discuss linear algebraic, combinatorial, and applied aspects of an exact model representation of binary arrays. As an illustration, we present two linear algebraic portraits of a string of characters.
The tools, ideas, and insights from linear algebra, abstract algebra, and functional analysis can be extremely useful to signal processing and system theory in various areas of engineering, science, and social science including…
In this paper it is shown that the compact linearization approach, that has been previously proposed only for binary quadratic problems with assignment constraints, can be generalized to arbitrary linear equations with positive coefficients…
Similar to how standard Young tableaux represent paths in the Young lattice, Latin rectangles may be use to enumerate paths in the poset of semi-magic squares with entries zero or one. The symmetries associated to determinant preserve this…
Latin squares are well studied combinatorial objects. In this paper we generalize the concept and propose new objects like Latin triangles, free Latin squares, Latin tetrahedra, free Latin cubes, etc. We start with a classic definition of…