Let R_{n} be the number of reduced Latin squares of order n.
We show that R_{n+1}=z_{n}=-2 mod n for prime n and
R_{n+1}=z_{n}=0 mod n for composite n. We provide a
congruence for z_{n} which we use to compute z_{n} mod
3 for all n≤60. Moreover, if n≥5 and n≠1 mod 3 then
z_{n}=0 mod 3 and if n=2×3^{k}+1 is prime, then
z_{n}=1 mod 3.

If κ is d-compound for all divisors d of n, then κ is
called compatible. If there is some integer polynomial f for which
κ(i)=f(i) mod n for all i, then κ is called a polynomial
orthomorphism. Let λ_{n} and π_{n} be the
number of canonical compatible and canonical polynomial
orthomorphisms, respectively. We find a formula for
λ_{n} and show that
λ_{n}=π_{n} if and only if
n=3^{a3} 5^{a5} p_{1}
p_{2}...p_{r} for distinct primes p_{i}≥7
and a_{3}≤3, a_{5}≤2. Finally we find some new
sufficient conditions for when a partial orthomorphism can be
completed to an orthomorphism.