The fundamental combinatorial structure of a net in CP^2 is its associated set of mutually orthogonal latin squares. We define equivalence classes of sets of orthogonal Latin squares by label equivalences of the lines of the corresponding net in CP^2. Then we count these equivalence classes for small cases. Finally, we prove that the realization spaces of these classes in CP^2 are empty to show some non-existence results for 4-nets in CP^2.
We give a list of canonical equivalence relations on discrete nets of the positive unit sphere of $c_0$. This generalizes results of W. T. Gowers and A. D. Taylor.
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 poset, and we completely describe the orbits, covering data, and maximal chains for squares of size 4, 5, and 6. The last item gives the number of Latin squares in these cases. To calculate efficiently for size 6, we in turn identify orbits with certain equivalence classes of hypergraphs.
In this paper we study pattern avoidance in Latin Squares, which gives us a two dimensional analogue of the well studied notion of pattern avoidance in permutations. Our main results include enumerating and characterizing the Latin Squares which avoid patterns of length three and a generalization of the Erdős-Szekeres theorem. We also discuss equivalence classes among longer patterns, and conclude by describing open questions of interest both in light of pattern avoidance and their potential to reveal information about the structure of Latin Squares. Along the way, we show that classical resul…
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 with conjugate symmetry and classify them according to several common notions of equivalence. We also do similar enumerations under additional hypotheses, such as assuming the Latin square is reduced, diagonal, idempotent or unipotent. Our data corrected an error in earlier literature and suggested several patterns that we then found proofs for, including (1…
We study the classes of $(u,m,{\bf e},s)$-nets and $(u,{\bf e},s)$-sequences, which are generalizations of $(u,m,s)$-nets and $(u,s)$-sequences, respectively. We show equivalence results that link the existence of $(u,m,{\bf e},s)$-nets and so-called mixed (ordered) orthogonal arrays, thereby generalizing earlier results by Lawrence, and Mullen and Schmid. We use this combinatorial equivalence principle to obtain new results on the possible parameter configurations of $(u,m,{\bf e},s)$-nets and $(u,{\bf e},s)$-sequences, which generalize in particular a result of Martin and Stinson.
We report the results of a computer investigation of sets of mutually orthogonal latin squares (MOLS) of small order. For $n\le9$ we 1. Determine the number of orthogonal mates for each species of latin square of order $n$. 2. Calculate the proportion of latin squares of order $n$ that have an orthogonal mate, and the expected number of mates when a square is chosen uniformly at random. 3. Classify all sets of MOLS of order $n$ up to various different notions of equivalence. We also provide a triple of latin squares of order 10 that is the closest to being a set of MOLS so far found.
A division sudoku is a latin square whose all six conjugates are sudoku squares. We enumerate division sudokus up to a suitable equivalence, introduce powerful invariants of division sudokus, and also study latin squares that are division sudokus with respect to multiple partitions at the same time. We use nearfields and affine geometry to construct division sudokus of prime power rank that are rich in sudoku partitions.
Orthogonal Latin Squares of Order Ten with Two Relations: A SAT Investigation
School of Computer Science, University of Waterloo（滑铁卢大学计算机学院）；Department of Mathematics and Statistics, University of Windsor（温莎大学数学与统计系）；School of Mathematics and Statistics, Carleton University（卡尔顿大学数学与统计学院）
Suppose that $Y_1,Y_2,Y_3$ are finite sets and $P\subseteq Y_1\times Y_2\times Y_3$. We say that $P$ embeds in a group $G$ if there exist injective maps $ϕ_i\colon Y_i\rightarrow G$ for $i=1,2,3$ such that $ϕ_1(y_1)ϕ_2(y_2)=ϕ_3(y_3)$ for each $(y_1,y_2,y_3)\in P$. Hirsch and Jackson asked for the cardinality of the smallest $P$ that embeds in some infinite group but not into any finite group. We prove that the answer to their question is 12. Moreover, we show that there are 50 examples of cardinality 12, up to equivalence, and each of them embeds in the (infinite) Baumslag group $G=\langle a,b…
Nielsen, Plotkin, and Winskel (1981) proved that every 1-safe Petri net $N$ unfolds into an event structure $\mathcal{E}_N$. By a result of Thiagarajan (1996 and 2002), these unfoldings are exactly the trace regular event structures. Thiagarajan (1996 and 2002) conjectured that regular event structures correspond exactly to trace regular event structures. In a recent paper (Chalopin and Chepoi, 2017, 2018), we disproved this conjecture, based on the striking bijection between domains of event structures, median graphs, and CAT(0) cube complexes. On the other hand, in Chalopin and Chepoi (2018)…
In this paper we study collections of mutually nearly orthogonal Latin squares ($\text{MNOLS}$), which come from a modification of the orthogonal condition for mutually orthogonal Latin squares. In particular, we find the maximum $μ$ such that there exists a set of $μ$ cyclic $\text{MNOLS}$ of order $n$ for $n \leq 18$, as well as providing a full enumeration of sets and lists of $μ$ cyclic $\text{MNOLS}$ of order $n$ under a variety of equivalences with $n \leq 18$. This resolves in the negative a conjecture that proposed the maximum $μ$ for which a set of $μ$ cyclic $\text{MNOLS}$ of order $…
