  The channel output entropy of a transmitted sequence is the entropy of the
possible channel outputs and similarly the channel input entropy of a received
sequence is the entropy of all possible transmitted sequences. The goal of this
work is to study these entropy values for the k-deletion, k-insertion channels,
where exactly k symbols are deleted, inserted in the transmitted sequence,
respectively. If all possible sequences are transmitted with the same
probability then studying the input and output entropies is equivalent. For
both the 1-deletion and 1-insertion channels, it is proved that among all
sequences with a fixed number of runs, the input entropy is minimized for
sequences with a skewed distribution of their run lengths and it is maximized
for sequences with a balanced distribution of their run lengths. Among our
results, we establish a conjecture by Atashpendar et al. which claims that for
the 1-deletion channel, the input entropy is maximized by the alternating
sequences over all binary sequences. This conjecture is also verified for the
2-deletion channel, where it is proved that constant sequences with a single
run minimize the input entropy.
