Enumeration formulas for latin and frequency squares
A frequency (quasi-latin) square is an \(n \times n\) array with entries \(1,2,\dots,m\) such that each number \(i\) occurs exactly \(\lambda_ i\) times in each row and column. Two frequency squares are isotopic if one can be obtained from the other by permuting rows, columns, and then numbers. The authors relate the number of latin squares of order \(n\) to the number of unordered 1-factorizations of reduced frequency squares from each isotopy class. Unfortunately, they note that the amount of computation needed to compute the number of latin squares of order 10 (the smallest unknown case) by these methods is still beyond the scope of currently available computers.
scientific article; zbMATH DE number 428310
scientific article