  For a fixed integer base $b\geq2$, we consider the number of compositions of
$1$ into a given number of powers of $b$ and, related, the maximum number of
representations a positive integer can have as an ordered sum of powers of $b$.
We study the asymptotic growth of those numbers and give precise asymptotic
formulae for them, thereby improving on earlier results of Molteni. Our
approach uses generating functions, which we obtain from infinite transfer
matrices. With the same techniques the distribution of the largest denominator
and the number of distinct parts are investigated.
