Divisors of the number of Latin rectangles

A k×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) . Let Rk,n be the number of reduced k × n Latin rectangles and m = bn/2c. We prove several results giving divisors of Rk,n. For example, (k − 1)! divides Rk,n when k ≤ m and m! divides Rk,n when m < k ≤ n. We establish a recurrence which determines the congruence class of Rk,n (mod t) for a range of different t. We use this to show that Rk,n ≡ ( (−1)k−1(k − 1)! )n−1 (mod n). In particular, this means that if n is prime, then Rk,n ≡ 1 (mod n) for 1 ≤ k ≤ n and if n is composite then Rk,n ≡ 0 (mod n) if and only if k is larger than the greatest prime divisor of n.