Gray codes for Fibonacci q-decreasing words
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An n-length binary word is q-decreasing, q≥ 1, if every of its length maximal factor of the form 0^a1^b satisfies a=0 or q· 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≥ 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.
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