  Zeckendorf proved that every positive integer has a unique partition as a sum
of non-consecutive Fibonacci numbers. We study the difference between the
number of summands in the partition of two consecutive integers. In particular,
let $L(n)$ be the number of summands in the partition of $n$. We characterize
all positive integers such that $L(n) > L(n+1)$, $L(n) < L(n+1)$, and $L(n) =
L(n+1)$. Furthermore, we call $n+1$ a peak of $L$ if $L(n) < L(n+1) > L(n+2)$
and a divot of $L$ if $L(n) > L(n+1) < L(n+2)$. We characterize all such peaks
and divots of $L$.
