Enumeration formulas for latin and frequency squares

Dénes, J.; Mullen, Gary L.

doi:10.1016/0012-365x(93)90152-j

Cited by 6 publications

(5 citation statements)

References 4 publications

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“…-De Gennaro [28] This misbelief highlights the need for this survey. Somehow a similar false claim was made by Mullen and Mummert in a 2007 book, despite Mullen (with Dénes) having already published a formula for L n in [33].…”

Section: -Erdős and Kaplansky [40]mentioning

confidence: 70%

“…, where s i (µ) is the number of copies of i in the partition µ. Theorem 6.2 will show that each L counted by X µ admits the same number of completions C µ to a Latin square. Dénes and Mullen [33] gave a formula for L n which is essentially…”

Section: -Erdős and Kaplansky [40]mentioning

confidence: 99%

“…Dénes and Mullen [33] also gave a more general formula, which can be interpreted as partitioning Latin squares into λ s × n Latin rectangles where n = λ 1 + λ 2 + • • • + λ m and counting the number of non-clashing replacements that can be made for each λ s × n Latin rectangle. Equation 9arises from the case when the partition of n is 2 + (n − 2).…”

Section: -Erdős and Kaplansky [40]mentioning

confidence: 99%

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The Many Formulae for the Number of Latin Rectangles

Stones¹

2010

Electron. J. Combin.

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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 Doyle's formula for $L_{k,n}$ is given in a closed form and is used to compute previously unpublished values of $L_{4,n}$, $L_{5,n}$ and $L_{6,n}$.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.

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Section: -Erdős and Kaplansky [40]mentioning

confidence: 70%

Section: -Erdős and Kaplansky [40]mentioning

confidence: 99%

Section: -Erdős and Kaplansky [40]mentioning

confidence: 99%

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The Many Formulae for the Number of Latin Rectangles

Stones¹

2010

Electron. J. Combin.

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“…It can also be shown that these designs are based on Latin squares, 20,21 and mutually orthogonal Latin squares exist only if the number of levels is a prime number or a power of a prime number, meaning that, whereas such designs exist for two, three, four and five levels, they do not exist for six levels, an important point to note when performing laboratory experiments. Another theorem is that a maximum of (N 2 1)/(l 2 1) mutually orthogonal factors are possible, i.e., six for five levels, five for four levels, and so on.…”

Section: Possible Orthogonal Designsmentioning

confidence: 99%

Multilevel Multifactor Designs for MultivariateCalibration

Brereton¹

1997

Analyst

254

123

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Multilevel experiments are common in chemistry, especially in calibration and mixture problems. This paper presents designs for l = 2-5 different concentration levels and l 2 corresponding experiments. The importance of orthogonality between successive factors is discussed. It is shown that up to 12 mutually orthogonal factors can be generated for a five-level design. Methods of generating designs are generalised. The designs are restricted to first-order (linear) models, typical of most instrumental calibration experiments.

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“…Thus grids G with f (W, G) = f (W) where W is a set of words can be viewed as a generalization of diagonal Latin squares. Much more can be said about Latin squares; see, for example, [4,5,6,8,10,11].…”

Section: Introductionmentioning

confidence: 99%