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On the number of Latin squares. (English) Zbl 1073.05013

Summary: 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 \(1 \leq k \leq n \leq 11\), (5) show that the proportion of Latin squares with a non-trivial symmetry group tends quickly to zero as the order increases.

MSC:

05B15 Orthogonal arrays, Latin squares, Room squares
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
15A15 Determinants, permanents, traces, other special matrix functions

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