This thesis primarily investigates the number Rk,n of reduced k X n Latin rectangles. Specifically, we find many congruences that involve Rk,n with the aim of improving our understanding of Rk,n. In general, the problem of finding Rk,n is difficult and furthermore, the literature contains many published errors. Modern enumeration algorithms, such as that of McKay and Wanless, require lengthy computations and storage of a large amount of data. Consequently, even into the future, the possibility of obtaining an erroneous result remains, for example, through a hardware or bookkeeping error. In this thesis we find many congruences satisfied by Rk,n so that future researchers will be able to check that their purported value of Rk,n satisfies these congruences. We extend the methodology developed in this thesis to encompass the number of certain graph factorisations, the number of orthomorphisms and partial orthomorphisms and the size of certain subsets of Latin hypercuboids. Consequently we find new congruences satisfied by all these numbers. Additionally, we give new sufficient conditions for when a partial orthomorphism admits a completion to an orthomorphism. In a 1997 paper, Drisko suggested some ideas for future research in the study of the Alon-Tarsi Conjecture, which we show to be futile. We find a new bound on the maximum size of an autotopism group of a Latin square which enables us to find new divisors of Rn,n for large n. A similar method gives a bound on the maximum number of k X k subsquares in a Latin square, for general k. Finally, we find new strong necessary conditions for when an isotopism can be an autotopism of some Latin square.
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