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arXiv · 2204.10069

Strings from linear recurrences and permutations: a Gray code

Elena Barcucci·Antonio Bernini·Renzo Pinzani
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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.

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Elena Barcucci, Antonio Bernini, Renzo Pinzani. 2022-04-21. Strings from linear recurrences and permutations: a Gray code. https://arxiv.org/abs/2204.10069

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