Consider the quantity:
for each index t of the string, let l(t) be the length of
the run to which the character pointed to by the index belongs
c(t) = (leftceil log(l(t)+1) rightceil + 1) / l(t)
What is E[c(t)] in terms of C_{rle}(w)?
(the expectation is taken over the uniform choice of an index t).
How would one estimate E[c(t)]?

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