## Abstract

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-1}z_{d}z ^{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.

Original language | English |
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Pages (from-to) | 277-289 |

Number of pages | 13 |

Journal | Finite Fields and their Applications |

Volume | 16 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jul 2010 |

Externally published | Yes |