Strings from linear recurrences and permutations: a Gray code

TL;DR

This paper explores the relationship between linear recurrence-based strings and permutations, establishing bijections and Gray codes for specific combinatorial objects related to generalized Fibonacci sequences.

Contribution

It introduces a bijection between certain binary strings avoiding runs of ones and permutations avoiding specific patterns, along with a Gray code for these permutations.

Findings

Bijection between strings and permutations avoiding certain patterns

Gray code for permutations with adjacent transpositions

Analysis of Fibonacci-based numeration systems

Abstract

Each positive increasing integer sequence $\{a_n\}_{n\geq 0}$ 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 $1^k$. We prove a bijection between the set %$F_n^{(k)}$ of strings of length $n$ and the set of permutations of $S_{n+1}(321,312,23\ldots(k+1)1)$. Finally, basing on a known Gray code for those strings, we define a Gray code for $S_{n+1}(321,312,23\ldots(k+1)1)$, where two consecutive permutations differ by an adjacent transposition.