Compound orthomorphisms of the cyclic group

An orthomorphism κ of Z_n is a permutation of Z_n such that i ← κ(i)-i is also a permutation. We say κ is canonical if κ(0) = 0 and define z_n to 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. An orthomorphism of Z_n is called compatible if it is d-compound for all divisors d of n. An orthomorphism κ of Z_n is called a polynomial orthomorphism if there exists an integer polynomial f such that κ(i) = f (i) (mod n) 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 t^d-1z_dz ^d_t canonical d-compound orthomorphisms of Z_n and each can be defined by d orthomorphisms of Z_t and one orthomorphism of Z_d. It is known that z_n = -2 (mod n) for prime n; we show that z_n = 0 (mod n) for composite n. We then deduce that R_n+1 = z_n (mod n) for all n, where R_n is the number of reduced Latin squares of order n. We find the value of z _n (mod 3) for (a) n ≤ 60, (b) n ≠ 1 (mod 3) and (c) when n is a prime of the form 2.3^k + 1. Let λ_n and π_n be the number of canonical compatible and canonical polynomial orthomorphisms, respectively. We give a formula for λ_n and find necessary and sufficient conditions for λ_n = π_n to hold. Finally, we find a new sufficient condition for when a partial orthomorphism can be completed to a d-compound orthomorphism.