On Additive Combinatorics of Permutations of ℤn.

Let ℤn denote the ring of integers modulo n. A permutation of ℤn is a sequence of n distinct elements of ℤn. Addition and subtraction of two permutations is defined element-wise. In this paper we consider two extremal problems on permutations of ℤn, namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that no sum of two distinct permutations in the collection is a permutation. Let the sizes be denoted by s(n) and t(n) respectively. The case when n is even is trivial in both the cases, with s(n) = 1 and t(n) = n!. For n odd, we prove (nφ(n))/2k ≤ s(n) ≤ n!·2-(n-1)/2/((n-1)/2)! and 2(n-1)/2·(n-1/2)! ≤ t(n) ≤ 2k · (n-1)!/φ(n), where k is the number of distinct prime divisors of n and φ is the Euler's totient function. [ABSTRACT FROM AUTHOR]