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 (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 ≤ k ≤ n ≤ 11, (5) show that the proportion of Latin squares with a non-trivial symmetry group tends quickly to zero as the order increases. Received September 3, 2004     

Difference Covering Arrays and Pseudo-Orthogonal Latin Squares

A pair of Latin squares, A and B, of order n, is said to be pseudo-orthogonal if each symbol in A is paired with every symbol in B precisely once, except for one symbol with which it is paired twice and one symbol with which it is not paired at all. A set of t Latin squares, of order n, are said to be mutually pseudo-orthogonal if they are pairwise pseudo-orthogonal. 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.     

Multidimensional Permanents and an Upper Bound on the Number of Transversals in Latin Squares

The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. A nonnegative matrix whose every 1‐dimensional plane sums to 1 is called polystochastic. A latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and each column. A transversal of such a square is a set of n entries such that no two entries share the same row, column, or symbol. Let T(n) be the maximum number of transversals over all latin squares of order n. Here, we prove that over the set of multidimensional polystochastic matrices of order n the permanent has a local extremum at the uniform matrix for whose every entry is equal to . Also, we obtain an asymptotic value of the maximal permanent for a certain set of nonnegative multidimensional matrices. In particular, we get that the maximal permanent of polystochastic matrices is asymptotically equal to the permanent of the uniform matrix, whence as a corollary we have an upper bound on the number of transversals in latin squares      

Latin squares with no small odd plexes

A k‐plex in a Latin square of order n is a selection of kn entries in which each row, column, and symbol is represented precisely k times. A transversal of a Latin square corresponds to the case k = 1. We show that for all even n > 2 there exists a Latin square of order n which has no k‐plex for any odd but does have a k‐plex for every other . © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 477–492, 2008     

Critical Sets of 2-Balanced Latin Rectangles

An (m, n, 2)-balanced Latin rectangle is an \({m \times n}\) array on symbols 0 and 1 such that each symbol occurs n times in each row and m times in each column, with each cell containing either two 0’s, two 1’s or both 0 and 1. We completely determine the structure of all critical sets of the full (m, n, 2)-balanced Latin rectangle (which contains 0 and 1 in each cell). If m, \({n \geq 2}\), the minimum size for such a structure is shown to be \({(m-1)(n-1)+1}\). Such critical sets in turn determine defining sets for (0, 1)-matrices.     

Latin squares with restricted transversals

We prove that for all odd m ≥ 3 there exists a latin square of order 3 m that contains an ( m ? 1 ) × m latin subrectangle consisting of entries not in any transversal. We prove that for all even n ≥ 10 there exists a latin square of order n in which there is at least one transversal, but all transversals coincide on a single entry. A corollary is a new proof of the existence of a latin square without an orthogonal mate, for all odd orders n ≥ 11 . Finally, we report on an extensive computational study of transversal‐free entries and sets of disjoint transversals in the latin squares of order n ? 9 . In particular, we count the number of species of each order that possess an orthogonal mate. © 2011 Wiley Periodicals, Inc. J Combin Designs 20:124‐141, 2012     

Partitioning 3-homogeneous latin bitrades

A latin bitrade is a pair of partial latin squares that define the difference between two arbitrary latin squares and of the same order. A 3-homogeneous bitrade has three entries in each row, three entries in each column, and each symbol appears three times in . Cavenagh [2] showed that any 3-homogeneous bitrade may be partitioned into three transversals. In this paper we provide an independent proof of Cavenagh’s result using geometric methods. In doing so we provide a framework for studying bitrades as tessellations in spherical, euclidean or hyperbolic space. Additionally, we show how latin bitrades are related to finite representations of certain triangle groups.      

Tail approximations to the density function in EVT

Let X ₁, X ₂, ...,X _n be independent identically distributed random variables with common distribution function F, which is in the max domain of attraction of an extreme value distribution, i.e., there exist sequences a _n > 0 and b _n ∈ ℝ such that the limit of exists. Assume the density function f (of F) exists. We obtain an uniformly weighted approximation to the tail density function f, and an uniformly weighted approximation to the tail density function of under some second order condition.Partially supported by a grant of the Swiss National Science Foundation.     

Existence of generalized Latin squares which are not embeddable in any group

Let n be a positive integer. A generalized Latin square of order n is an \(n\times n\) matrix such that the elements in each row and each column are distinct. In this paper, we show that for any integer \(n\ge 6\) and any integer m where \(m\in \left\{ n, n+1, \dots , \frac{n(n+1)}{2}-2\right\} \), there exists a commutative generalized Latin square of order n with m distinct elements which is not embeddable in any group. In addition, we show that for any integer \(r\ge 3\) and any integer s where \(s\in \{ r, r+1, \dots , r^2-2\}\), there exists a non-commutative generalized Latin square of order r with s distinct elements which is not embeddable in any group.     

Degree Conditions for Matchability in 3‐Partite Hypergraphs

We study conjectures relating degree conditions in 3‐partite hypergraphs to the matching number of the hypergraph, and use topological methods to prove special cases. In particular, we prove a strong version of a theorem of Drisko [14] (as generalized by the first two authors [2]), that every family of $2n-1$ matchings of size n in a bipartite graph has a partial rainbow matching of size n. We show that milder restrictions on the sizes of the matchings suffice. Another result that is strengthened is a theorem of Cameron and Wanless [11], that every $n\times n$ Latin square has a generalized diagonal (set of n entries, each in a different row and column) in which no symbol appears more than twice. We show that the same is true under the weaker condition that the square is row‐Latin.     

Line Transversals to Disjoint Balls

We prove that the set of directions of lines intersecting three disjoint balls in ℝ³ in a given order is a strictly convex subset of . We then generalize this result to n disjoint balls in ℝ ^d . As a consequence, we can improve upon several old and new results on line transversals to disjoint balls in arbitrary dimension, such as bounds on the number of connected components and Helly-type theorems.     

On the Vertices of the d-Dimensional Birkhoff Polytope

Let us denote by Ω _n the Birkhoff polytope of n×n doubly stochastic matrices. As the Birkhoff–von Neumann theorem famously states, the vertex set of Ω _n coincides with the set of all n×n permutation matrices. Here we consider a higher-dimensional analog of this basic fact. Let $\varOmega^{(2)}_{n}$ be the polytope which consists of all tristochastic arrays of order n. These are n×n×n arrays with nonnegative entries in which every line sums to 1. What can be said about $\varOmega ^{(2)}_{n}$ ’s vertex set? It is well known that an order-n Latin square may be viewed as a tristochastic array where every line contains n?1 zeros and a single 1 entry. Indeed, every Latin square of order n is a vertex of $\varOmega^{(2)}_{n}$ , but as we show, such vertices constitute only a vanishingly small subset of $\varOmega^{(2)}_{n}$ ’s vertex set. More concretely, we show that the number of vertices of $\varOmega ^{(2)}_{n}$ is at least $(L_{n})^{\frac{3}{2}-o(1)}$ , where L _n is the number of order-n Latin squares. We also briefly consider similar problems concerning the polytope of n×n×n arrays where the entries in every coordinate hyperplane sum to 1, improving a result from Kravtsov (Cybern. Syst. Anal., 43(1):25–33, 2007). Several open questions are presented as well.     

Exact Distribution of the Local Score for Markovian Sequences

Let be a sequence of letters taken in a finite alphabet Θ. Let be a scoring function and the corresponding score sequence where X _i = s(A _i ). The local score is defined as follows: . We provide the exact distribution of the local score in random sequences in several models. We will first consider a Markov model on the score sequence , and then on the letter sequence . The exact P-value of the local score obtained with both models are compared thanks to several datasets. They are also compared with previous results using the independent model.     

Transversals and multicolored matchings

Ryser conjectured that the number of transversals of a latin square of order n is congruent to n modulo 2. Balasubramanian has shown that the number of transversals of a latin square of even order is even. A 1‐factor of a latin square of order n is a set of n cells no two from the same row or the same column. We prove that for any latin square of order n, the number of 1‐factors with exactly n ? 1 distinct symbols is even. Also we prove that if the complete graph K_2n, n ≥ 8, is edge colored such that each color appears on at most edges, then there exists a multicolored perfect matching. © 2004 Wiley Periodicals, Inc.     

Maximum transversal in partial Latin squares and rainbow matchings

In a partial Latin square P a set of distinct entries, such that no two of which are in the same row or column is called a transversal. By the size of a transversal T, we mean the number of its entries. We define a duplex to be a partial Latin square of order n containing 2n entries such that exactly two entries lie in each row and column and each of n symbols occurs exactly twice. We show that determining the maximum size of a transversal in a given duplex is an NP-complete problem. This problem relates to independent sets in certain subfamilies of cubic graphs. Generalizing the concept of transversals in edge coloring of graphs we are led to introduce the concept of rainbow matching. We show that if each color appears at most twice then it is a polynomial time problem to know whether there exists a rainbow matching of size at least ⌊n/2⌋-t for each fixed t, where n is the order of the graph. As an application we show that for any fixed t, there is a polynomial time algorithm which decides whether α(G)?n-t, for any graph G on 2n vertices containing a perfect matching. At the end we mention some other applications of rainbow matching.     

An Anti-Ramsey Theorem on Diamonds

Let G be the diamond (the graph obtained from K ₄ by deleting an edge) and, for every n ≥ 4, let f(n, G) be the minimum integer k such that, for every edge-coloring of the complete graph of order n which uses exactly k colors, there is at least one copy of G all whose edges have different colors. Let ext(n, {C ₃, C ₄}) be the maximum number of edges of a graph on n vertices free of triangles and squares. Here we prove that for every n ≥ 4,

Dirac’s Type Sufficient Conditions for Hamiltonicity and Pancyclicity

Let G = (V, E) be a any simple, undirected graph on n ≥ 3 vertices with the degree sequence . We consider the class of graphs satisfying the condition where , is a positive integer. It is known that is hamiltonian if θ ≤ δ. In this paper,

Bounds on eigenvalues of Dirichlet Laplacian

In this paper, we investigate an eigenvalue problem of Dirichlet Laplacian on a bounded domain Ω in an n-dimensional Euclidean space R ⁿ . If λ _k+1 is the (k + 1)th eigenvalue of Dirichlet Laplacian on Ω, then, we prove that, for n ≥ 41 and and, for any n and with , where j _p,k denotes the k-th positive zero of the standard Bessel function J _p (x) of the first kind of order p. From the asymptotic formula of Weyl and the partial solution of the conjecture of Pólya, we know that our estimates are optimal in the sense of order of k.Q.-M. Cheng was partially Supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of ScienceH. Yang was partially Supported by Chinese NSF, SF of CAS and NSF of USA     

Gradient Estimates for the Heat Semigroup on H-Type Groups

By utilizing the Poincaré inequality and representation formulae, it is shown that on the Heisenberg type group, ℍ(2n, m), there exists a constant C > 0 such that