Bijective BWT based compression schemes

We show that for any string $w$ of length $n$, $r_B = O(z\log^2 n)$, where $r_B$ and $z$ are respectively the number of character runs in the bijective Burrows-Wheeler transform (BBWT), and the number of Lempel-Ziv 77 factors of $w$. We can further induce a bidirectional macro scheme of size $O(r_B)$ from the BBWT. Finally, there exists a family of strings with $r_B = \Omega(\log n)$ but having only $r=2$ character runs in the standard Burrows--Wheeler transform (BWT). However, a lower bound for $r$ is the minimal run-length of the BBWTs applied to the cyclic shifts of $w$, whose time complexity might be $o(n^2)$ in the light that we show how to compute the Lyndon factorization of all cyclic rotations in $O(n)$ time. Considering also the rotation operation performing cyclic shifts, we conjecture that we can transform two strings having the same Parikh vector to each other by BBWT and rotation operations, and prove this conjecture for the case of binary alphabets and permutations.