A Congruence Connecting Latin Rectangles and Partial Orthomorphisms

Let χ(n, d) be the number of injective maps σ : S → Zn \ {0} such that (a) S ⊂ Zn of cardinality |S | = n − d, (b) σ(i) , i for all i ∈ S and (c) σ(i) − i . σ( j) − j (mod n) whenever i , j. Let Rk,n be the number of k × n reduced Latin rectangles. We show that Rk,n ≡ χ(p, n − p) (n − p)!(n − p − 1)!2 (n − k)! Rk−p,n−p (mod p) when p is a prime and n ≥ k ≥ p+1. This allows us to calculate explicit congruences for Rn,n for n ≤ 31. We show that χ(n, d) is divisible by d2/ gcd(n, d) when 1 ≤ d < n and establish several formulae for χ(n, n − a). In particular, for each a there exists μa such that, on each congruence class modulo μa, χ(n, n − a) is determined by a polynomial in n.