Compound orthomorphisms of the cyclic group

An orthomorphism κ of Z_n is a permutation such that i -> κ(i)-i is also a permutation. We say κ is canonical if κ(0)=0 and let z_n be the number of canonical orthomorphisms of Z_n. If n=dt and κ(i)=κ(j) mod d whenever i=j mod d then κ is called d-compound. There are exactly t^d-1 z_d z_t^d canonical d-compound orthomorphisms of Z_n and each can be defined by d orthomorphisms of Z_t and one orthomorphism of Z_d.

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 p₁ p₂...p_r for distinct primes p_i≥7 and a₃≤3, a₅≤2. Finally we find some new sufficient conditions for when a partial orthomorphism can be completed to an orthomorphism.