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# On Additive Combinatorics of Permutations of Z(n)

Chandran, Sunil L and Rajendraprasad, Deepak and Singh, Nitin (2014) On Additive Combinatorics of Permutations of Z(n). In: DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE, 16 (2). pp. 35-40. PDF dis_mat_the_com_sci_16-2_35_2014.pdf - Published Version Restricted to Registered users only Download (299kB) | Request a copy
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## Abstract

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 (n phi(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)!/phi(n), where k is the number of distinct prime divisors of n and phi is the Euler's totient function.

Item Type: Journal Article DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE DISCRETE MATHEMATICS THEORETICAL COMPUTER SCIENCE Copy right for this article belongs to the DISCRETE MATHEMATICS THEORETICAL COMPUTER SCIENCE, 62 RUE DU CARDINAL MATHIEU, F-54000 NANCY, FRANCE. sums of permutations; orthomorphisms; reverse free families Division of Electrical Sciences > Computer Science & AutomationDivision of Physical & Mathematical Sciences > Mathematics 08 Nov 2014 05:32 08 Nov 2014 05:32 http://eprints.iisc.ac.in/id/eprint/50207 View Item