<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>A permutation <italic>p</italic> of [<italic>k</italic>] = {1, 2, 3, …, <italic>k</italic>} is called Layman permutation iff <italic>i</italic> + <italic>p</italic>(<italic>i</italic>) is a Fibonacci number for 1 ≤ <italic>i</italic> ≤ <italic>k</italic>. This concept is introduced by Layman in the A097082 entry of the Encyclopedia of Integers Sequences, that is the number of Layman permutations of [<italic>n</italic>]. In this paper, we will study Layman permutations. We introduce the notion of the Fibonacci complement of a natural number, that plays a crucial role in our investigation. Using this notion we prove some results on the number of Layman permutations, related to a conjecture of Layman that is implicit in the A097083 entry of OEIS.</p>
</abstract>

A permutation p of [k] = {1, 2, 3, …, k} is called Layman permutation iff i + p(i) is a Fibonacci number for 1 ≤ i ≤ k. This concept is introduced by Layman in the A097082 entry of the Encyclopedia of Integers Sequences, that is the number of Layman permutations of [n]. In this paper, we will study Layman permutations. We introduce the notion of the Fibonacci complement of a natural number, that plays a crucial role in our investigation. Using this notion we prove some results on the number of Layman permutations, related to a conjecture of Layman that is implicit in the A097083 entry of OEIS.

A short note on Layman permutations

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"A short note on Layman permutations"}

A permutation p of [k] = {1, 2, 3, …, k} is called Layman permutation iff i + p(i) is a Fibonacci number for 1 ≤ i ≤ k. This concept is introduced by Layman in...

A short note on Layman permutations

Published Online: Jan 19, 2023

Page range: 231 - 238

Received: Dec 20, 2021

DOI: https://doi.org/10.2478/ausm-2022-0015

Keywords
permutations, Fibonacci numbers, unique perfect matching

© 2022 Péter Hajnal, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

A short note on Layman permutations

Published Online: Jan 19, 2023

Page range: 231 - 238

Received: Dec 20, 2021

DOI: https://doi.org/10.2478/ausm-2022-0015

Keywordspermutations, Fibonacci numbers, unique perfect matching

© 2022 Péter Hajnal, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
permutations, Fibonacci numbers, unique perfect matching