Integrals, partitions, and cellular automata

Abstract

We prove that ∫ 0 1 − log ⁡ f ( x ) x d x = π 2 3 a b , \begin{equation*}\int _0^1\frac {-\log f(x)}xdx=\frac {\pi ^2}{3ab},\end{equation*} where f ( x ) f(x) is the decreasing function that satisfies f a − f b = x a − x b f^a-f^b=x^a-x^b , for 0 > a > b 0>a>b . When a a is an integer and b = a + 1 b=a+1 we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having a a consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.