Rectangles in latin squares

Colouring games based on autotopisms of Latin hyper-rectangles

Every partial colouring of a Hamming graph is uniquely related to a partial Latin hyper-rectangle. In this paper we introduce the $Θ$-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 $Θ$ 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.

随机拉丁方与拉丁矩中的子方

Subsquares in random Latin squares and rectangles

该研究分析随机拉丁矩与拉丁方的子结构问题，先证明随机k×n拉丁矩含特定稀疏部分拉丁矩的概率公式，再据此得出n阶随机拉丁方渐近几乎必然无大于$c\sqrt{n\log n}$阶拉丁子方的结论，推进了随机拉丁方的子方存在性研究。

Lattice path enumeration for semi-magic squares by Latin rectangles

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.

Automatic Counting of Generalized Latin Rectangles and Trapezoids

In this case study in ``fully automated enumeration'', we illustrate how to take full advantage of symbolic computation by developing (what we call) `symbolic-dynamical-programming' algorithms for computing many terms of `hard to compute sequences', namely the number of Latin trapezoids, generalized derangements, and generalized three-rowed Latin rectangles. At the end we also sketch the proof of a generalization of Ira Gessel's 1987 theorem that says that for any number of rows, k, the number of Latin rectangles with k rows and n columns is P-recursive in n. Our algorithms are fully implement…

A computational algebraic geometry approach to classify partial Latin rectangles

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…

基于λ元集合的(3×n)阶拉丁矩形数量的一个简单证明

A simple proof on the number of $(3 \times n)$-Latin rectangles based on a set of $λ$ elements

针对已有(3×n)阶拉丁矩形色多项式证明方法复杂的问题，研究者采用数学归纳思想与恰当着色方法，给出了该结论的更简单证明。

Computing autotopism groups of partial Latin rectangles: a pilot study

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 $2 \times 2$ s…

随机拉丁矩形中的子方阵

Subsquares in random Latin rectangles

本文证明了均匀随机 $k\times n$ 拉丁矩形以高概率不含阶数4及以上的真拉丁子方阵，验证了相关猜想，并确定了阶数2和3子方阵的期望数量。

Enumeration and classification of self-orthogonal partial Latin rectangles by using the polynomial method

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 Gröbner 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…

关于简单多重拉丁矩的Ryser定理

Ryser's Theorem for Simple Multi-Latin Rectangle

本文推广了拉丁矩的补全理论，证明了在预先指定符号出现次数且单元格无重复符号的设定下，简单多重拉丁矩可扩展为拉丁方，并证实了Cavenagh等人关于部分拉丁方嵌入的猜想。

Enumerating partial Latin rectangles

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 $\mathr…

轮廓矩形、分配与拉丁杨图

Outline Rectangles, Allocations, and Latin Young Diagrams

本文引入杨图的分配概念，借助Hilton定理证明杨图存在分配等价于其为拉丁杨图，据此证明了Chow等人提出的宽杨图猜想对含三种不同行长度的杨图成立。

Hall's Condition for Partial Latin Squares

Hall's Condition is a necessary condition for a partial latin square to be completable. Hilton and Johnson showed that for a partial latin square whose filled cells form a rectangle, Hall's Condition is equivalent to Ryser's Condition, which is a necessary and sufficient condition for completability. We give what could be regarded as an extension of Ryser's Theorem, by showing that for a partial latin square whose filled cells form a rectangle, where there is at most one empty cell in each column of the rectangle, Hall's Condition is a necessary and sufficient condition for completability. It …

An analogue of Ryser's Theorem for partial Sudoku squares

In 1956 Ryser gave a necessary and sufficient condition for a partial latin rectangle to be completable to a latin square. In 1990 Hilton and Johnson showed that Ryser's condition could be reformulated in terms of Hall's Condition for partial latin squares. Thus Ryser's Theorem can be interpreted as saying that any partial latin rectangle $R$ can be completed if and only if $R$ satisfies Hall's Condition for partial latin squares. We define Hall's Condition for partial Sudoku squares and show that Hall's Condition for partial Sudoku squares gives a criterion for the completion of partial Sudok…

Ryser's Theorem for $ρ$-latin Rectangles

Let $L$ be an $n\times n$ array whose top left $r\times s$ subarray is filled with $k$ different symbols, each occurring at most once in each row and at most once in each column. We find necessary and sufficient conditions that ensure the remaining cells of $L$ can be filled such that each symbol occurs at most once in each row and at most once in each column, and each symbol occurs a prescribed number of times in $L$. The case where the prescribed number of times each symbol occurs is $n$ was solved by Ryser (Proc. Amer. Math. Soc. 2 (1951), 550--552), and the case $s=n$ was settled by Goldwa…

A Matroid Generalization of a Result on Row-Latin Rectangles

Let A be an m \times n matrix in which the entries of each row are all distinct. Drisko showed that, if m \ge 2n-1, then A has a transversal: a set of n distinct entries with no two in the same row or column. We generalize this to matrices with entries in a matroid. For such a matrix A, we show that if each row of A forms an independent set, then we can require the transversal to be independent as well. We determine the complexity of an algorithm based on the proof of this result. Lastly, we observe that m \ge 2n-1 appears to force the existence of not merely one but many transversals. We disc…

Triples of Orthogonal Latin and Youden Rectangles For Small Orders

We have performed a complete enumeration of non-isotopic triples of mutually orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$. Here we will present a census of such triples, classified by various properties, including the order of the autotopism group of the triple. As part of this we have also achieved the first enumeration of pairwise orthogonal triples of Youden rectangles. We have also studied orthogonal triples of $k \times 8$ rectangles which are formed by extending mutually orthogonal triples with non-trivial autotopisms one row at a time, and requiring that the autotopism g…

相互正交拉丁矩形集合的枚举

Enumeration of Sets of Mutually Orthogonal Latin Rectangles

本文枚举 k≤n≤7 时非同痕的 t 个相互正交 k×n 拉丁矩形集合，定义并追踪齐次、传递及逐步传递结构，并据此说明除 Hughes 平面外阶数至多 9 的射影平面均可由逐步传递 MOLR 构造。

Sudoku Rectangle Completion

Over the last decade, Sudoku, a combinatorial number-placement puzzle, has become a favorite pastimes of many all around the world. In this puzzle, the task is to complete a partially filled $9 \times 9$ square with numbers 1 through 9, subject to the constraint that each number must appear once in each row, each column, and each of the nine $3 \times 3$ blocks. Sudoku squares can be considered a subclass of the well-studied class of Latin squares. In this paper, we study natural extensions of a classical result on Latin square completion to Sudoku squares. Furthermore, we use the procedure de…

Completing partial Latin squares with two filled rows and three filled columns

Consider a partial Latin square $P$ where the first two rows and first three columns are completely filled, and every other cell of $P$ is empty. It has been conjectured that all such partial Latin squares of order at least $8$ are completable. Based on a technique by Kuhl and McGinn we describe a framework for completing partial Latin squares in this class. Moreover, we use our method for proving that all partial Latin squares from this family, where the intersection of the nonempty rows and columns form a Latin rectangle with three distinct symbols, is completable.

Some existence problems regarding partial Latin squares

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 Latin square and with applications to Latin squares is whether a partial Latin square with a given set of conditions exists. The goal of this article is to introduce some problems of this kind and answer some existence questions regarding partial Latin squares

Latin squares, partial latin squares and its generalized quotients

A (partial) Latin square is a table of multiplication of a (partial) quasigroup. Multiplication of a (partial) quasigroup may be considered as a set of triples. We give a necessary and sufficient condition when a set of triples is a quotient of a (partial) Latin square.

嵌入均衡矩形

Embedding Equitable Rectangles

研究部分拉丁方阵的补全问题，推广Ryser定理到更一般情形，给出填充剩余单元格使各符号出现次数均衡的充要条件，统一并推广了多个经典结果。

Multi-latin squares

A multi-latin square of order $n$ and index $k$ is an $n\times n$ array of multisets, each of cardinality $k$, such that each symbol from a fixed set of size $n$ occurs $k$ times in each row and $k$ times in each column. A multi-latin square of index $k$ is also referred to as a $k$-latin square. A $1$-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square. In this note we show that any partially filled-in $k$-latin square of order $m$ embeds in a $k$-latin square of order $n$, for each $n\geq 2m$, thus generalizing Evans' …

避免给定拉丁方的同构类的部分拉丁方

Avoiding a partial Latin square with an isotope of a given Latin square

该研究针对足够大的n阶拉丁方，通过仅置换行和列的方法，证明了可使其避开任意指定的n阶部分拉丁方，证明结合了Hall定理与随机置换的初等估计。

On the number of Latin squares

We (1) determine the number of Latin rectangles with 11 columns and each possible number of rows, including the Latin squares of order~11, (2) answer some questions of Alter by showing that the number of reduced Latin squares of order $n$ is divisible by $f!$ where $f$ is a particular integer close to $\frac12n$, (3) provide a formula for the number of Latin squares in terms of permanents of $(+1,-1)$-matrices, (4) find the extremal values for the number of 1-factorisations of $k$-regular bipartite graphs on $2n$ vertices whenever $1\leq k\leq n\leq11$, (5) show that the proportion of Latin sq…

Embedding partial Latin squares in Latin squares with many mutually orthogonal mates

We show that any partial Latin square of order $n$ can be embedded in a Latin square of order at most $16n^2$ which has at least $2n$ mutually orthogonal mates. We also show that for any $t\geq 2$, a pair of orthogonal partial Latin squares of order $n$ can be embedded into a set of $t$ mutually orthogonal Latin squares (MOLS) of order a polynomial with respect to $n$. Furthermore, the constructions that we provide show that MOLS($n^2$)$\geq$MOLS($n$)+2, consequently we give a set of $9$ MOLS($576$). The maximum known size of a set of MOLS($576$) was previously given as $8$ in the literature.

On the chromatic number of Latin square graphs

The chromatic number of a Latin square is the least number of partial transversals which cover its cells. This is just the chromatic number of its associated Latin square graph. Although Latin square graphs have been widely studied as strongly regular graphs, their chromatic numbers appear to be unexplored. We determine the chromatic number of a circulant Latin square, and find bounds for some other classes of Latin squares. With a computer, we find the chromatic number for all main classes of Latin squares of order at most eight.

Erdős问题725的次线性范围内拉丁方的Godsil–McKay渐近公式

The Godsil--McKay Asymptotic for Latin Rectangles in the Sublinear Range of Erdős Problem 725

本研究部分解决了Erdős问题725，将k×n拉丁矩形数量的Godsil–McKay渐近公式的适用范围从k=o(n^{6/7})拓展至所有k=o(n)，给出了带绝对常数的对数误差界，且结果已在Lean中形式化验证。

Recent results on Choi's orthogonal Latin squares

Choi Seok-Jeong studied Latin squares at least 60 years earlier than Euler although this was less known. He introduced a pair of orthogonal Latin squares of order 9 in his book. Interestingly, his two orthogonal non-double-diagonal Latin squares produce a magic square of order 9, whose theoretical reason was not studied. There have been a few studies on Choi's Latin squares of order 9. The most recent one is Ko-Wei Lih's construction of Choi's Latin squares of order 9 based on the two $3 \times 3$ orthogonal Latin squares. In this paper, we give a new generalization of Choi's orthogonal Latin …

Enumeration of Latin squares with conjugate symmetry

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…

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 $Θ$, the cardinality of the set $\mathcal{PLS}_Θ$ of partial Latin squares which are invariant under $Θ$ 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 $Θ$ determines the possible sizes of the elements of $\mathcal{PLS}_…

Mutually orthogonal Sudoku Latin squares and their graphs

We introduce a graph attached to mutually orthogonal Sudoku Latin squares. The spectra of the graphs obtained from finite fields are explicitly determined. As a corollary, we then use the eigenvalues to distinguish non-isomorphic Sudoku Latin squares.

Quasi-transversal in Latin Squares

In this paper, we first present the relation between a transversal in a Latin square with some concepts in its Latin square graph, and give an equivalent condition for a Latin square has an orthogonal mate. The most famous open problem involving Combinatorics is to find maximum number of disjoint transversals in a Latin square. So finding some family of decomposable Latin squares into disjoint transversals is our next aim. In the next section, we give an equivalent statement of a conjecture which has been attributed to Brualdi, Stein and Ryser by the concept of quasi-transversal. Finally, we p…

There are asymptotically the same number of Latin squares of each parity

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 between the numbers of reduced Latin squares of each possible parity as the order $n\rightarrow\infty$.

Completions of epsilon-dense partial Latin squares

A classical question in combinatorics is the following: given a partial latin square P, when can we complete P to a latin square L? In this paper, we will investigate the class of \leqε-dense partial latin squares: partial latin squares in which each symbol, row, and column contains \leqεn-many nonblank cells. A conjecture of Nash-Williams on triangulations of graphs led Daykin and Häggkvist to conjecture that all \leq(1/4)-dense partial latin squares are completable. In this paper, we will discuss the proof methods and results used in previous attempts to resolve this conjecture, introduce a …

On the distances between Latin squares and the smallest defining set size

In this note we show that for each Latin square $L$ of order $n\geq 2$, there exists a Latin square $L'\neq L$ of order $n$ such that $L$ and $L'$ differ in at most $8\sqrt{n}$ cells. Equivalently, each Latin square of order $n$ contains a Latin trade of size at most $8\sqrt{n}$. We also show that the size of the smallest defining set in a Latin square is $Ω(n^{3/2})$. %That is, there are constants $c$ and $n_0$ such that for any $n>n_0$ the size of the smallest defining %set of order $n$ is at least $cn^{3/2}$.

A polynomial embedding of pairs of orthogonal partial latin squares

We show that a pair of orthogonal partial latin squares of order $n$ can be embedded in a pair of orthogonal latin squares of order at most $16n^4$ and all orders greater than or equal to $48n^4$. This paper provides the first direct polynomial order embedding construction in the literature.

On the number of transversals in latin squares

The logarithm of the maximum number of transversals over all latin squares of order $n$ is greater than $\frac{n}{6}(\ln n+ O(1))$.

Parity Types, Cycle Structures and Autotopisms of Latin Squares

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 for the size of autotopy groups. A fast algorithm for finding the autotopy group of a Latin square, based on the cycle decomposition of its rows, is presented.

Intercalates and Discrepancy in Random Latin Squares

An intercalate in a Latin square is a $2\times2$ Latin subsquare. Let $N$ be the number of intercalates in a uniformly random $n\times n$ Latin square. We prove that asymptotically almost surely $N\ge\left(1-o\left(1\right)\right)\,n^{2}/4$, and that $\mathbb{E}N\le\left(1+o\left(1\right)\right)\,n^{2}/2$ (therefore asymptotically almost surely $N\le fn^{2}$ for any $f\to\infty$). This significantly improves the previous best lower and upper bounds. We also give an upper tail bound for the number of intercalates in two fixed rows of a random Latin square. In addition, we discuss a problem of L…

Equitable partitions of Latin-square graphs

We study equitable partitions of Latin-square graphs, and give a complete classification of those whose quotient matrix does not have an eigenvalue $-3$.

Domination for latin square graphs

In combinatorics, a latin square is a $n\times n$ matrix filled with n different symbols, each occurring exactly once in each row and exactly once in each column. Associated to each latin square, we can define a simple graph called a latin square graph. In this article, we compute lower and upper bounds for the domination number and the k-tuple total domination numbers of such graphs. Moreover, we describe a formula for the 2-tuple total domination number.

Mutually orthogonal latin squares with large holes

Two latin squares are orthogonal if, when they are superimposed, every ordered pair of symbols appears exactly once. This definition extends naturally to `incomplete' latin squares each having a hole on the same rows, columns, and symbols. If an incomplete latin square of order $n$ has a hole of order $m$, then it is an easy observation that $n \ge 2m$. More generally, if a set of $t$ incomplete mutually orthogonal latin squares of order $n$ have a common hole of order $m$, then $n \ge (t+1)m$. In this article, we prove such sets of incomplete squares exist for all $n,m \gg 0$ satisfying $n \g…

Difference Covering Arrays and Pseudo-Orthogonal Latin Squares

Difference arrays are used in applications such as software testing, authentication codes and data compression. Pseudo-orthogonal Latin squares are used in experimental designs. A special class of pseudo-orthogonal Latin squares are the mutually nearly orthogonal Latin squares (MNOLS) first discussed in 2002, with general constructions given in 2007. In this paper we develop row complete MNOLS from difference covering arrays. We will use this connection to settle the spectrum question for sets of 3 mutually pseudo-orthogonal Latin squares of even order, for all but the order 146.

Hamilton transversals in random Latin squares

Gyárfás and Sárközy conjectured that every $n\times n$ Latin square has a `cycle-free' partial transversal of size $n-2$. We confirm this conjecture in a strong sense for almost all Latin squares, by showing that as $n \rightarrow \infty$, all but a vanishing proportion of $n\times n$ Latin squares have a Hamilton transversal, i.e. a full transversal for which any proper subset is cycle-free. In fact, we prove a counting result that in almost all Latin squares, the number of Hamilton transversals is essentially that of Taranenko's upper bound on the number of full transversals. This result str…

On completing three cyclic transversals to a latin square

Let $P$ be a partial latin square of prime order $p>7$ consisting of three cyclically generated transversals. Specifically, let $P$ be a partial latin square of the form: \[ P=\{(i,c+i,s+i),(i,c'+i,s'+i),(i,c''+i,s''+i)\mid 0 \leq i< p\} \] for some distinct $c,c',c''$ and some distinct $s,s',s''$. In this paper we show that any such $P$ completes to a latin square which is diagonally cyclic.

广义拉丁矩形中的成对反射对称性

Pairwise Reflection Symmetry in Generalized Latin Rectangles

研究广义拉丁矩形中成对反射对称性的存在性，通过构造和整数规划发现，列重复度λ=1当且仅当n为2的幂，且奇数λ时存在性更微妙。

Distance in Latin Squares

A Latin square of order $n$ is an $n\times n$ array which contains $n$ distinct symbols exactly once in each row and column. We define the adjacent distance between two adjacent cells (containing integers) to be their difference modulo $n$, and inner distance of a Latin square to be the minimum of adjacent distances in the Latin square. By first establishing upper bounds and then constructing squares with said inner distance, we found the maximum inner distance of an $n \times n$ Latin square to be $\left\lfloor\frac{n-1}{2}\right\rfloor$. We then studied special kinds of Latin squares such as…

On uniformly generating Latin squares

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 distribution (approximately). The central issue is the construction of "moves" that connect the squares. Most of their lengthy paper is to prove that the associated graph is indeed connected. We give a short proof of this fact by using the concepts of Latin bitrades.

Completing Partial Latin Squares - Alternative Proof

The problem of completing a partially specified n by n Latin square is solved by an alternative proof, based on filling the rows (or diagonals) from 1 to n, using an extended form of Hall's marriage theorem.

具有最大基数的对称量子拉丁方

Symmetric Quantum Latin Squares with Maximal Cardinality

本文证明了所有$n\geq 6$阶存在最大基数的对称量子拉丁方，且所有$4k+2$阶存在最大基数的对称幂等量子拉丁方。

quasigroups和拉丁方的自paratopism

Autoparatopisms of Quasigroups and Latin Squares

研究quasigroups和拉丁方的自paratopism性质，确定n≤17时Par(n)的结构，并分析n趋于无穷时Par(n)在所有paratopism中的比例。

The number of Latin squares of order 11

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) 6108088657705958932053657 isomorphism classes of one-factorizations of $K_{11,11}$; (iii) 12216177315369229261482540 isotopy classes of Latin squares of order 11; (iv) 1478157455158044452849321016 isomorphism classes of loops of order 11; and (v) 19464657391668924966791023043937578299025 isomorphism classes of quasigroups of order 11. The enumeration is constructive for the 1151666…