Permutations from an arithmetic setting
Abstract
Let $m, n$ be positive integers such that $m>1$ divides $n$. In this paper, we introduce a special class of piecewiseaffine permutations of the finite set $[1, n]:=\{1, \ldots, n\}$ with the property that the reduction $\pmod m$ of $m$ consecutive elements in any of its cycles is, up to a cyclic shift, a fixed permutation of $[1, m]$. Our main result provides the cycle decomposition of such permutations. We further show that such permutations give rise to permutations of finite fields. In particular, we explicitly obtain classes of permutation polynomials of finite fields whose cycle decomposition and its inverse are explicitly given.
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 arXiv:
 arXiv:1904.12920
 Bibcode:
 2019arXiv190412920R
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Combinatorics;
 05A05;
 11B50;
 11T22
 EPrint:
 15 pages. To appear in Discrete Mathematics