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Divisors of the number of Latin rectangles

A k by n Latin rectangle on the symbols {1, 2,...,n} is called reduced
if the first row is (1,2,...,n) and the first column is
(1,2,...,k)^{T}. Let R_{k,n} be the number of k by n
reduced Latin rectangles and m = floor(n/2). We prove several results
giving divisors of R_{k,n}. For example, (k-1)! divides
R_{k,n} when k ≤ m and m! divides R_{k,n} when m <
k ≤ n. We establish a recurrence which determines the congruence
of R_{k,n} mod t for a range of different t. We use this to
show that R_{k,n} = ((-1)^{k-1}(k-1)!)^{n-1}
mod n. In particular, this means that if n is prime, then
R_{k,n} = 1 mod n for 1 ≤ k ≤ n and if n is composite
then R_{k,n} = 0 mod n if and only if k is greater than the
largest prime divisor of n.
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