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- W4293395838 abstract "Let $mathbb{Z}_n$ denote the ring of integers modulo $n$. In this paper we consider two extremal problems on permutations of $mathbb{Z}_n$, 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 the sum of any two distinct permutations in the collection is not 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 $s(n)geq (nphi(n))/2^k$ where $k$ is the number of distinct prime divisors of $n$. When $n$ is an odd prime we prove $s(n)leq frac{e^2}{pi} n ((n-1)/e)^frac{n-1}{2}$. For the second problem, we prove $2^{(n-1)/2}.(frac{n-1}{2})!leq t(n)leq 2^k.(n-1)!/phi(n)$ when $n$ is odd." @default.
- W4293395838 created "2022-08-28" @default.
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- W4293395838 date "2014-02-17" @default.
- W4293395838 modified "2023-09-27" @default.
- W4293395838 title "On Additive Combinatorics of Permutations of mathbb{Z}_n" @default.
- W4293395838 doi "https://doi.org/10.48550/arxiv.1402.3970" @default.
- W4293395838 hasPublicationYear "2014" @default.
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