On the number of Latin squares

Details On the number of Latin squares On the number of Latin squares

2009-09-11

arxiv.org/abs/0909.2101v1

B. D. McKay and I. M. Wanless, On the number of Latin squares, Ann. Combin. 9 (2005) 335-344

Mathematics

Combinatorics

11 pages

Scientific paper

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.

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