Strings from linear recurrences and permutations: A Gray code

Each positive increasing integer sequence {an}n≥0 can serve as a numeration system to represent each non-negative by means of suitable coefficient strings. We analyse the case k-generalized Fibonacci sequences leading binary strings avoiding 1k. prove bijection between set such length n and Sn+1(321,312,23…(k+1)1) which is subset symmetric group Sn+1 permutations n+1 permutation patterns 321,312, 234…(k+1)1, where 234…(k+1) stands for integers from 2 k+1. Finally, basing on known Gray code those strings, we define Sn+1(321,312,234…(k+1)1), two consecutive differ an adjacent transposition.