A Classical Diophantine Problem and Modular Forms of Weight 3/2.

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

 $$\frac{1}{2}n,$$
 (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.

https://users.monash.edu.au/~iwanless/papers/LS11AC.pdf

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 squares with a non-trivial symmetry group tends quickly to zero as the order increases.

Congruences for Catalan and Motzkin numbers and related sequences

A $k \times n$ Latin rectangle $L$ is a $k \times n$ array, with symbols from a set of cardinality $n$, such that each row and each column contains only distinct symbols. If $k=n$ then $L$ is a Latin square. Let $L_{k,n}$ be the number of $k \times n$ Latin rectangles. We survey (a) the many combinatorial objects equivalent to Latin squares, (b) the known bounds on $L_{k,n}$ and approximations for $L_n$, (c) congruences satisfied by $L_{k,n}$ and (d) the many published formulae for $L_{k,n}$ and related numbers. We also describe in detail the method of Sade in finding $L_{7,7}$, an important milestone in the enumeration of Latin squares, but which was privately published in French. Doyle's formula for $L_{k,n}$ is given in a closed form and is used to compute previously unpublished values of $L_{4,n}$, $L_{5,n}$ and $L_{6,n}$. We reproduce the three formulae for $L_{k,n}$ by Fu that were published in Chinese. We give a formula for $L_{k,n}$ that contains, as special cases, formulae of (a) Fu, (b) Shao and Wei and (c) McKay and Wanless. We also introduce a new equation for $L_{k,n}$ whose complexity lies in computing subgraphs of the rook's graph.

/pdf/the-many-formulae-for-the-number-of-latin-rectangles-ds291z7x4v.pdf

The Many Formulae for the Number of Latin Rectangles

Prime and prime power divisibility of Catalan numbers

How Many Latin Squares are There

Uniquely intersectable graphs

An integer a is called a congruent number if and only if there are positive integer solutions to the system of equations x2 + ay2 = z2 and x2 - ay2 = t2. In this note congruent numbers are discussed and a table of known square-free congruent numbers less than 1000 is exhibited. 1. Introduction. An integer a is called a congruent number if and only if there exist positive integer solutions to the rationalized system of Diophantine equations (1) x2 + ay2= Z2 and x2 - ay2 t2. It is easy to see that cn2 is a congruent number if and only if c is a congruent number. Thus it suffices to study only square-free congruent numbers. The earliest reference to congruent numbers in the literature dates back to a tenth century Arab manuscript. For a more detailed account of the early history of con- gruent numbers, see Dickson (3, Chapter 16, pp. 459-472). More recent references to congruent numbers can be found in several texts. See, for example, Uspensky and

/pdf/a-note-on-congruent-numbers-3vjj1y01s7.pdf

A Note on Congruent Numbers

In this paper applications of results obtained by these authors for a generalized version of a problem proposed by Smirnov, are considered. The areas of application explored are system interface, queueing, transportation flow, and sequential analysis. The included table should be invaluable to the reader in applying these results. Finally the relationship between the limiting and exact expressions relating to this table is also explored.

Ronald Alter

Papers

Prime and prime power divisibility of Catalan numbers

How Many Latin Squares are There

Uniquely intersectable graphs

A Note on Congruent Numbers

Applications of a generalized combinatorial problem of smirnov