Abstract
A length n binary word is q-decreasing, q⩾1, if every of its maximal factors of the form 0a1b satisfies a=0 or q⋅a>b. In particular, in 1-decreasing words every run of 0s is immediately followed by a strictly shorter run of 1s. We show constructively that these words are in bijection with binary words having no occurrences of 1q+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 1q+1 in terms of the frequency of 1-bits. In the second part of our paper, we provide an efficient exhaustive generating algorithm for q-decreasing words in lexicographic order, for any q⩾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 Eğecioğlu and Iršič.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
