  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.
