On additive combinatorics of permutations of ℤ&lt;sub&gt;n&lt;/sub&gt;

L Sunil Chandran,Nitin Singh,Deepak Rajendraprasad

doi:10.46298/dmtcs.2074

Abstract

Special issue PRIMA 2013 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.

Highlights

For n ∈ Z, let Zn denote the ring {0, . . . , n − 1} with + and . as addition and multiplication modulo n respectively
We are interested in obtaining bounds on the maximum size of a subset P of S(Zn) in the case when two distinct permutations in P sum up to a permutation, and in the case when no two distinct permutations in P sum up to a permutation
We consider the maximum sizes of collections of permutations of Zn under two different constraints, namely, (i) Sum of any two distinct permutations in the collection is again a permutation

Summary

Introduction

For n ∈ Z, let Zn denote the ring {0, . . . , n − 1} with + and . as addition and multiplication modulo n respectively. For n ∈ Z, let Zn denote the ring {0, . Let S(Zn) denote the set of all permutations of the set Zn. We are interested in obtaining bounds on the maximum size of a subset P of S(Zn) in the case when two distinct permutations in P sum up to a permutation, and in the case when no two distinct permutations in P sum up to a permutation. As far as we know the problems considered above are new, though a similar looking problem for difference of permutations is well studied, in the form of mutually orthogonal orthomorphisms of finite groups. 1365–8050 c 2014 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

Preliminaries

Results and Proofs

Differences of Permutations being Permutation

Latin Squares