  We prove that $$\int_0^1\frac{-\log f(x)}xdx=\frac{\pi^2}{3ab}$$ where $f(x)$
is the decreasing function that satisfies $f^a-f^b=x^a-x^b$, for $0<a<b$. When
$a$ is an integer and $b=a+1$ we deduce several combinatorial results. These
include an asymptotic formula for the number of integer partitions not having
$a$ consecutive parts, and a formula for the metastability thresholds of a
class of threshold growth cellular automaton models related to bootstrap
percolation.
