Compound orthomorphisms of the cyclic group

An orthomorphism κ of Zn is a permutation such that i -> κ(i)-i is also a permutation. We say κ is canonical if κ(0)=0 and let zn be the number of canonical orthomorphisms of Zn. If n=dt and κ(i)=κ(j) mod d whenever i=j mod d then κ is called d-compound. There are exactly td-1 zd ztd canonical d-compound orthomorphisms of Zn and each can be defined by d orthomorphisms of Zt and one orthomorphism of Zd.

Let Rn be the number of reduced Latin squares of order n. We show that Rn+1=zn=-2 mod n for prime n and Rn+1=zn=0 mod n for composite n. We provide a congruence for zn which we use to compute zn mod 3 for all n≤60. Moreover, if n≥5 and n≠1 mod 3 then zn=0 mod 3 and if n=2×3k+1 is prime, then zn=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 λn and πn be the number of canonical compatible and canonical polynomial orthomorphisms, respectively. We find a formula for λn and show that λnn if and only if n=3a3 5a5 p1 for distinct primes pi≥7 and a3≤3, a5≤2. Finally we find some new sufficient conditions for when a partial orthomorphism can be completed to an orthomorphism.