The number of distinct latin squares as a group-theoretical constant

Jucys, A. A.

doi:10.1016/0097-3165(76)90020-0

Cited by 6 publications

(3 citation statements)

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“…which has been rearranged and a problem corrected -the last equation in [49] should have f n(n−r)+k instead of f n(n−r) . Jucys [71] constructed an algebra A n over C, with the "magic squares" as a basis, which were actually n × n non-negative integer matrices with row and column sums equal to n. Multiplication in A n was defined using a "structure constant," which, in one case, was L n . An isomorphism was identified between A n and a subalgebra of the group algebra of the symmetric group S n 2 over C. Representation theory was then used to give an expression for L n in terms of eigenvalues of a particular element of A n .…”

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 Doyle's formula for $L_{k,n}$ is given in a closed form and is used to compute previously unpublished values of $L_{4,n}$, $L_{5,n}$ and $L_{6,n}$.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: 99%

“…-Jucys [71] Light Jr. [86] (see also [85]) gave an equation for the number of "truncated Latin rectangles" which, for Latin rectangles, simplifies to…”

Section: -Erdős and Kaplansky [40]mentioning

confidence: 99%

The Many Formulae for the Number of Latin Rectangles

Stones¹

2010

Electron. J. Combin.

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“…General formulae for L n have been found by MacMahon [27,28] (see also [41]), Jucys [21], Light Jr. [26], Nechvatal [33,34], Gessel [16], Shao and Wei [39], Fu [14], Denés and Mullen [6] and McKay and Wanless [31], however they are all impractical for enumeration purposes. Kuznetsov [24] (see also [30]) provided estimates for L n for n 20.…”

Section: Introductionmentioning

confidence: 96%

Divisors of the number of Latin rectangles

Stones

Wanless

2010

Journal of Combinatorial Theory, Series A

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A k × n Latin rectangle on the symbols {1, 2, . . . ,n} is called reduced if the first row is (1, 2, . . . ,n) and the first column is (1, 2, . . . ,k) T . Let R k,n be the number of reduced k × n Latin rectangles and m = n/2 . We prove several results giving divisors of R k,n . For example, (k − 1)! divides R k,n when k m and m! divides R k,n when m < k n. We establish a recurrence which determines the congruence class of R k,n (mod t) for a range of different t. We use this to show that R k,n ≡ ((−1) k−1 (k − 1)!) n−1 (mod n). In particular, this means that if n is prime, then R k,n ≡ 1 (mod n) for 1 k n and if n is composite then R k,n ≡ 0 (mod n) if and only if k is larger than the greatest prime divisor of n.

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Loops, Latin Squares and Strongly Regular Graphs: An Algorithmic Approach via Algebraic Combinatorics

Heinze

Klin

2009

Algorithmic Algebraic Combinatorics and Gröbner Bases

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The number of distinct latin squares as a group-theoretical constant

Cited by 6 publications

References 6 publications

The Many Formulae for the Number of Latin Rectangles

The Many Formulae for the Number of Latin Rectangles

Divisors of the number of Latin rectangles

Loops, Latin Squares and Strongly Regular Graphs: An Algorithmic Approach via Algebraic Combinatorics

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