  A lumping of a Markov chain is a coordinate-wise projection of the chain. We
characterise the entropy rate preservation of a lumping of an aperiodic and
irreducible Markov chain on a finite state space by the random growth rate of
the cardinality of the realisable preimage of a finite-length trajectory of the
lumped chain and by the information needed to reconstruct original trajectories
from their lumped images. Both are purely combinatorial criteria, depending
only on the transition graph of the Markov chain and the lumping function. A
lumping is strongly k-lumpable, iff the lumped process is a k-th order Markov
chain for each starting distribution of the original Markov chain. We
characterise strong k-lumpability via tightness of stationary entropic bounds.
In the sparse setting, we give sufficient conditions on the lumping to both
preserve the entropy rate and be strongly k-lumpable.
