On additive combinatorics of permutations of ℤ_n

L. Sunil Chandran (1) , Deepak Rajendraprasad (2) , Nitin Singh (3)

1 CSA - Department of Computer Science and Automation [Bangalore]
2 Department of Computer Science (Haifa)
3 Department of Mathematics (Bangalore)

Let ℤ_n denote the ring of integers modulo n. A permutation of ℤ_n is a sequence of n distinct elements of ℤ_n. Addition and subtraction of two permutations is defined element-wise. In this paper we consider two extremal problems on permutations of ℤ_n, 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^k≤s(n)≤n!· 2^-(n-1)/2((n-1)/2)! and 2^(n-1)/2·(n-1 / 2)!≤t(n)≤ 2^k·(n-1)!/φ(n), where k is the number of distinct prime divisors of n and φ is the Euler's totient function.

Mots clés: reverse free families; orthomorphisms; sums of permutations

Domaines: Mathématique discrète [cs.DM]; Interface homme-machine [cs.HC]

L. Sunil Chandran, Deepak Rajendraprasad, Nitin Singh. On additive combinatorics of permutations of ℤ_n. Discrete Mathematics and Theoretical Computer Science, 2014, Vol. 16 no. 2 (2), pp.35--40. ⟨10.46298/dmtcs.2074⟩. ⟨hal-01185614⟩