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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 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 Id for Latest eprints 08 Nov 2014 05:32 08 Nov 2014 05:32 http://eprints.iisc.ac.in/id/eprint/50207 View Item