Gray codes for Fibonacci q-decreasing words

TL;DR

This paper introduces a new class of binary words called q-decreasing words, establishes their combinatorial properties, and develops efficient algorithms for their enumeration and Gray code generation, including solutions to existing conjectures.

Contribution

It provides a bijection between q-decreasing words and Fibonacci-numbered binary words, and constructs Gray codes for these words, including a special case settling a recent conjecture.

Findings

q-decreasing words are enumerated by generalized Fibonacci numbers.

Efficient algorithms for generating q-decreasing words in lexicographic order.

Existence of 3-Gray codes and a specific 1-Gray code for 1-decreasing words.

Abstract

An $n$-length binary word is $q$-decreasing, $q\geq 1$, if every of its length maximal factor of the form $0^a1^b$ satisfies $a=0$ or $q\cdot a > b$.We show constructively that these words are in bijection with binary words having no occurrences of $1^{q+1}$, and thus they are enumerated by the $(q+1)$-generalized Fibonacci numbers. We give some enumerative results and reveal similarities between $q$-decreasing words and binary words having no occurrences of $1^{q+1}$ in terms of frequency of $1$ bit. In the second part of our paper, we provide an efficient exhaustive generating algorithm for $q$-decreasing words in lexicographic order, for any $q\geq 1$, show the existence of 3-Gray codes and explain how a generating algorithm for these Gray codes can be obtained. Moreover, we give the construction of a more restrictive 1-Gray code for $1$-decreasing words, which in particular settles a conjecture stated recently in the context of interconnection networks by E\u{g}ecio\u{g}lu and Ir\v{s}i\v{c}.