Integrals, Partitions, and Cellular Automata

Holroyd, Alexander E.; Liggett, Thomas M.; Romik, Dan

Integrals, Partitions, and Cellular Automata

Authors

Holroyd, Alexander E.

Liggett, Thomas M.

Romik, Dan

Description

We prove that $$\int_0^1\frac{-\log f(x)}xdx=\frac{π^2}{3ab}$$ 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.
Revised version. 28 pages, 2 figures