Row-Hamiltonian Latin squares and Falconer varieties

A \emph{Latin square} is a matrix of symbols such that each symbol occurs exactly once in each row and column. A Latin square $L$ is \emph{row-Hamiltonian} if the permutation induced by each pair of distinct rows of $L$ is a full cycle permutation. Row-Hamiltonian Latin squares are equivalent to perfect $1$-factorisations of complete bipartite graphs. For the first time, we exhibit a family of Latin squares that are row-Hamiltonian and also achieve precisely one of the related properties of being column-Hamiltonian or symbol-Hamiltonian. This family allows us to construct non-trivial, anti-associative, isotopically $L$-closed loop varieties, solving an open problem posed by Falconer in 1970

Cycle structures of autotopisms of the Latin squares of order up to 11

The cycle structure of a Latin square autotopism Θ = (α, β, γ) is the triple (lα, lβ, lγ), where lδ is the cycle structure of δ, for all δ ∈ {α, β, γ}. In this paper we study some properties of these cycle structures and, as a consequence, we give a classification of all autotopisms of the Latin squares of order up to 11

Orthogonal trades in complete sets of MOLS

Let Bₚ be the Latin square given by the addition table for the integers modulo an odd prime p (i.e. the Cayley table for (Zₚ, +)). Here we consider the properties of Latin trades in Bₚ which preserve orthogonality with one of the p−1 MOLS given by the finite field construction. We show that for certain choices of the orthogonal mate, there is a lower bound logarithmic in p for the number of times each symbol occurs in such a trade, with an overall lower bound of (log p)² / log log p for the size of such a trade. Such trades imply the existence of orthomorphisms of the cyclic group which differ from a linear orthomorphism by a small amount. We also show that any transversal in Bₚ hits the main diagonal either p or at most p − log₂ p – 1 times. Finally, if p ≡ 1 (mod 6) we show the existence of a Latin square which is orthogonal to Bₚ and which contains a 2 × 2 subsquare

Additive triples of bijections, or the toroidal semiqueens problem

We prove an asymptotic for the number of additive triples of bijections $\{1,\dots,n\}\to\mathbb{Z}/n\mathbb{Z}$, that is, the number of pairs of bijections $\pi_1,\pi_2\colon \{1,\dots,n\}\to\mathbb{Z}/n\mathbb{Z}$ such that the pointwise sum $\pi_1+\pi_2$ is also a bijection. This problem is equivalent to counting the number of orthomorphisms or complete mappings of $\mathbb{Z}/n\mathbb{Z}$, to counting the number of arrangements of $n$ mutually nonattacking semiqueens on an $n\times n$ toroidal chessboard, and to counting the number of transversals in a cyclic Latin square. The method of proof is a version of the Hardy--Littlewood circle method from analytic number theory, adapted to the group $(\mathbb{Z}/n\mathbb{Z})^n$.Comment: 22 page

The set of autotopisms of partial Latin squares

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 invariant under $\Theta$ only depends on the conjugacy class of the latter, or, equivalently, on its cycle structure. In the current paper, the cycle structures of the set of autotopisms of partial Latin squares are characterized and several related properties studied. It is also seen that the cycle structure of $\Theta$ determines the possible sizes of the elements of $\mathcal{PLS}_{\Theta}$ and the number of those partial Latin squares of this set with a given size. Finally, it is generalized the traditional notion of partial Latin square completable to a Latin square.Comment: 20 pages, 4 table

Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups

An $r \times s$ partial Latin rectangle $(l_{ij})$ is an $r \times s$ matrix containing elements of $\{1,2,\ldots,n\} \cup \{\cdot\}$ such that each row and each column contain at most one copy of any symbol in $\{1,2,\ldots,n\}$. An entry is a triple $(i,j,l_{ij})$ with $l_{ij} \neq \cdot$. Partial Latin rectangles are operated on by permuting the rows, columns, and symbols, and by uniformly permuting the coordinates of the set of entries. The stabilizers under these operations are called the autotopism group and the autoparatopism group, respectively. We develop the theory of symmetries of partial Latin rectangles, introducing the concept of a partial Latin rectangle graph. We give constructions of $m$-entry partial Latin rectangles with trivial autotopism groups for all possible autoparatopism groups (up to isomorphism) when: (a) $r=s=n$, i.e., partial Latin squares, (b) $r=2$ and $s=n$, and (c) $r=2$ and $s \neq n$