Universal updates of Dyck-nest signatures

Let $0<k\in\mathbb{Z}$. The anchored Dyck words of length $n=2k+1$ (obtained by prefixing a 0-bit to each Dyck word of length $2k$ and used to reinterpret the Hamilton cycles in the odd graph $O_k$ and the middle-levels graph $M_k$ found by M\"utze et al.) represent in $O_k$ (resp., $M_k$) the cycles of an $n$- (resp., $2n$-) 2-factor and its cyclic (resp., dihedral) vertex classes, and are equivalent to Dyck-nest signatures. A sequence is obtained by updating these signatures according to the depth-first order of a tree of restricted growth strings (RGS's), reducing the RGS-generation of Dyck words by collapsing to a single update the time-consuming $i$-nested castling used to reach each non-root Dyck word or Dyck nest. This update is universal, for it does not depend on $k$.