The middle-levels graph Mkβ (0<kβZ) has a dihedral quotient
pseudograph Rkβ whose vertices are the k-edge ordered trees T, each T
encoded as a (2k+1)-string F(T) formed via βDFS by: {\bf(i)}
(βBFS-assigned) Kierstead-Trotter lexical colors 0,β¦,k for
the descending nodes; {\bf(ii)} asterisks β for the k ascending edges. Two
ways of corresponding a restricted-growth k-string Ξ± to each T
exist, namely one Stanley's way and a novel way that assigns F(T) to Ξ±
via nested substring-swaps. These swaps permit to sort V(Rkβ) as an ordered
tree that allows a lexical visualization of Mkβ as well as the Hamilton
cycles of Mkβ constructed by P. Gregor, T. M\"utze and J. Nummenpalo.Comment: 26 pages, 8 figures, 10 table