Strings from linear recurrences and permutations: A Gray code

Abstract

Each positive increasing integer sequence {an}n≥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 1k. We prove a bijection between the set of such strings of length n and the set Sn+1(321,312,23…(k+1)1) which is a subset of the symmetric group Sn+1 of the permutations of length n+1 avoiding the permutation patterns 321,312, and 234…(k+1)1, where 234…(k+1) stands for the sequence of increasing integers from 2 to k+1.Finally, basing on a known Gray code for those strings, we define a Gray code for Sn+1(321,312,234…(k+1)1), where two consecutive permutations differ by an adjacent transposition.