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