Strings from linear recurrences and permutations: A Gray code

• Published in: Theoretical computer science Vol. 938; pp. 112 - 120
• Main Authors: Barcucci, Elena, Bernini, Antonio, Pinzani, Renzo
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 26.11.2022
• Subjects: Gray code; k-generalized Fibonacci sequences; Numeration systems; Pattern avoiding permutations; Numeration systems; Gray code; k-generalized Fibonacci sequences; Pattern avoiding permutations
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2022.10.012

Each positive increasing integer sequence {an}n≥0 can serve as a numeration system to represent each non-negative integer by means of suitable coefficient strings. We analyse the case of k-generalized Fibonacci sequences leading to the binary strings avoiding 1k. We prove a bijection between the set of such strings of length n and the set Sn+1(321,312,23…(k+1)1) which is a subset of the symmetric group Sn+1 of the permutations of length n+1 avoiding the permutation patterns 321,312, and 234…(k+1)1, where 234…(k+1) stands for the sequence of increasing integers from 2 to k+1. Finally, basing on a known Gray code for those strings, we define a Gray code for Sn+1(321,312,234…(k+1)1), where two consecutive permutations differ by an adjacent transposition.