https://hal-univ-bourgogne.archives-ouvertes.fr/hal-03114503Baril, Jean-LucJean-LucBarilLIB - Laboratoire d'Informatique de Bourgogne [Dijon] - UB - Université de BourgogneKirgizov, SergeySergeyKirgizovLIB - Laboratoire d'Informatique de Bourgogne [Dijon] - UB - Université de BourgogneVajnovszki, VincentVincentVajnovszkiLIB - Laboratoire d'Informatique de Bourgogne [Dijon] - UB - Université de BourgogneGray codes for Fibonacci q-decreasing wordsHAL CCSD2022[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO][INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Kirgizov, Sergey2021-01-19 00:01:482022-08-04 17:07:382021-01-19 00:01:48enJournal articles1An $n$-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}.