## A Congruence Connecting Latin Rectangles and Partial Orthomorphisms

Let χ(n,d) be the number of injective maps σ : S
-> Z_{n} \ {0} such that
(a) S is a subset of Z_{n} and has cardinality |S|=n-d,

(b) σ(i) ≠ i for all i in S and

(c) σ(i)-i ≠ σ(j)-j mod n for
distinct i,j in S.

Let R_{k,n} be the number of k × n
reduced Latin rectangles. We show that

R_{k,n} = χ(p,n-p)
(n-p)! (n-p-1)!^{2}
R_{k-p,n-p}/(n-k)! mod p

when p is a
prime and n ≥ k ≥ p+1. This allows us to calculate explicit
congruences for R_{n,n} for n ≤ 31. We show that χ(n,d)
is divisible by d^{2}/gcd(n,d) when 1 ≤ d < n and establish
several formulae for χ(n,n-a). In particular, for each a
there exists μ_{a} such that, on each congruence class modulo
μ_{a}, χ(n,n-a) is determined by a polynomial in n.