Related papers: Autoparatopisms of Quasigroups and Latin Squares
A paratopism is an action on a Latin hypercube of dimension d and order n which is an element of the wreath product $S_n \wr S_{d+1}$. A paratopism is said to be an autoparatopism if there is at least one Latin hypercube which is mapped to…
The parity type of a Latin square is defined in terms of the numbers of even and odd rows and columns. It is related to an Alon-Tarsi-like conjecture that applies to Latin squares of odd order. Parity types are used to derive upper bounds…
An autotopism of a Latin square is a triple $(\alpha,\beta,\gamma)$ of permutations such that the Latin square is mapped to itself by permuting its rows by $\alpha$, columns by $\beta$, and symbols by $\gamma$. Let $\mathrm{Atp}(n)$ be the…
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…
Symmetries of a partial Latin square are determined by its autotopism group. Analogously to the case of Latin squares, given an isotopism $\Theta$, the cardinality of the set $\mathcal{PLS}_{\Theta}$ of partial Latin squares which are…
There exists a bijection between the set of Latin squares of order $n$ and the set of feasible solutions of the 3-dimensional planar assignment problem ($3PAP_n$). In this paper, we prove that, given a Latin square isotopism $\Theta$, we…
This paper will present some intuitive interpretation of the parastrophe transformations of arbitrary Latin square. With this trick, we can generate the parastrophes of arbitrary Latin square directly from the original one without…
Latin squares have been historically used in order to create statistical designs in which, starting from a small number of experiments, it can be obtained a large experimental space. In this sense, the optimization of the selection of Latin…
An algorithm that uses the cycle structure of the rows, or the columns, of a Latin square to compute its autotopy group is introduced. As a result, a bound for the size of the autotopy group is obtained. This bound is used to show that the…
A Latin square is reduced if its first row and column are in natural order. For Latin squares of a particular order $n$ there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality…
In a Latin square, every row can be interpreted as a permutation, and therefore has a parity (even or odd). We prove that in a uniformly random $n\times n$ Latin square, the $n$ row parities are very well approximated by a sequence of $n$…
A quasigroup is a pair $(Q, \cdot)$ where $Q$ is a non-empty set and $\cdot$ is a binary operation on $Q$ such that for every $(u, v) \in Q^2$ there exists a unique $(x, y) \in Q^2$ such that $u \cdot x = v = y \cdot u$. Let $q$ be an odd…
A Latin square has six conjugate Latin squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a Latin square has conjugate symmetry if at least two of its six conjugates are equal. We enumerate Latin squares…
The set LS(n) of Latin squares of order $n$ can be represented in $\mathbb{R}^{n^3}$ as a $(n-1)^3$-dimensional 0/1-polytope. Given an autotopism $\Theta=(\alpha,\beta,\gamma)\in\mathfrak{A}_n$, we study in this paper the 0/1-polytope…
The $n\times n$ doubly stochastic matrices constitute a polytope in $\mathbb{R}^{n^2}$, and by Birkhoff's theorem, its vertex set coincides with the set of order-$n$ permutation matrices.\\ A tristochastic array is an $n \times n\times n$…
Constructive and nonconstructive techniques are employed to enumerate Latin squares and related objects. It is established that there are (i) 2036029552582883134196099 main classes of Latin squares of order 11; (ii)…
A latin square of order $n$ is an $n\times n$ array of $n$ symbols in which each symbol occurs exactly once in each row and column. A transversal of such a square is a set of $n$ entries such that no two entries share the same row, column…
A Latin square of order $n$ with symbols $a_1,\ldots,a_n$ can be considered as a multiplication table for binary operation in the set $A=\{a_1,\ldots,a_n\}$. We prove that, if this operation is associative, then $A$ is a group.
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…
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…