Iterated quotient equals a power quotient up to permutation

For every $e,k\ge 1$, the $e^k$-quotient $Q_{e^k}(\lambda )$ is a permutation of level $k$ of the $e$-quotient tower. Writing a runner residue modulo $e^k$ in base $e$ shows that taking one quotient chooses one digit at a time; iteration may reverse the order of those digits but selects the same runner partitions.