Compound orthomorphisms of the cyclic group

An orthomorphism κ of Zn is a permutation of Zn such that i↦κ(i)−i is also a permutation. We say κ is canonical if κ(0)=0 and define zn to be the number of canonical orthomorphisms of Zn. If n=dt and whenever then κ is called d-compound. An orthomorphism of Zn is called compatible if it is d-compound for all divisors d of n. An orthomorphism κ of Zn is called a polynomial orthomorphism if there exists an integer polynomial f such that for all i. We develop the theory of compound, compatible and polynomial orthomorphisms and the relationships between these classes.We show that there are exactly canonical d-compound orthomorphisms of Zn and each can be defined by d orthomorphisms of Zt and one orthomorphism of Zd. It is known that for prime n; we show that for composite n. We then deduce that for all n, where Rn is the number of reduced Latin squares of order n. We find the value of for (a) n⩽60, (b) and (c) when n is a prime of the form 2⋅k3+1. Let λn and πn be the number of canonical compatible and canonical polynomial orthomorphisms, respectively. We give a formula for λn and find necessary and sufficient conditions for λn=πn to hold. Finally, we find a new sufficient condition for when a partial orthomorphism can be completed to a d-compound orthomorphism.