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))/2 ≤ s(n) ≤ n!·2 −(n−1)/2 ((n−1)/2)! and 2 (n−1)/2 · (n−1 2 )! ≤ t(n) ≤ 2 · (n− 1)!/φ(n), where k is the number of distinct prime divisors of n and φ is the Euler’s totient function.