On the Number of Latin Squares

• DOI: 10.1007/s00026-005-0261-7
• Publication Date: 2005-09-30T22:26:15Z
• Authors: Brendan D. McKay; Ian M. Wanless
• Keywords: Keywords: 1-factorisation; 05B15; Enumeration; Latin square; FOS: Mathematics; Permanent; Mathematics - Combinatorics; Regular bipartite graph; Combinatorics (math.CO); 0101 mathematics; 01 natural sciences

11 pages<br/>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.<br/>