text: In this paper we propose an algorithm for enumerating diagonal Latin squares of small order. It relies on specific properties of diagonal Latin squares to employ symmetry breaking techniques, and on several heuristic optimizations and bit arithmetic techniques to make use of computational power of state-of-the-art CPUs. Using this approach we enumerated diagonal Latin squares of order at most 9, and vertically symmetric diagonal Latin squares of order at most 10.
  @cite_6: 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 ( 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.
  @cite_12: On a @math chessboard, the maximum number of nonattacking kings that can be placed is @math , since each @math cell can have at most one king. Let @math denote the number of ways that @math nonattacking kings can be placed on a @math chessboard. The purpose of this paper is to prove the following result. Theorem. For each @math there are constants @math , @math , and $0
  @cite_8: 
summary: Authors are not aware of algorithms developed specifically for enumeration of diagonal Latin squares. Papers @cite_8 @cite_12 @cite_6 describe the approaches that led to enumeration of Latin squares of orders 9, 10 and 11. The corresponding algorithms heavily rely on the ability to permute rows and columns to construct equivalence classes and evaluate their properties. To the best of our knowledge the results of these papers are not applicable to diagonal Latin squares because the vast majority of row-column permutations break diagonal property. Also, diagonal Latin squares form relatively small equivalence classes.