Annals of Combinatorics

, Volume 9, Issue 3, pp 335–344

On the Number of Latin Squares

Original Paper

DOI: 10.1007/s00026-005-0261-7

Cite this article as:
McKay, B.D. & Wanless, I.M. Ann. Comb. (2005) 9: 335. doi:10.1007/s00026-005-0261-7


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

AMS Subject Classification.



Latin squareenumeration1-factorisationpermanentregular bipartite graph

Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  1. 1.Department of Computer ScienceAustralian National UniversityCanberraAustralia
  2. 2.School of Engineering and LogisticsCharles Darwin UniversityDarwinAustralia