Integrals, partitions, and cellular automata

Holroyd, Alexander; Liggett, Thomas; Romik, Dan

doi:10.1090/S0002-9947-03-03417-2

Integrals, partitions, and cellular automata
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by Alexander E. Holroyd, Thomas M. Liggett and Dan Romik PDF

Trans. Amer. Math. Soc. 356 (2004), 3349-3368 Request permission

Abstract:

We prove that \begin{equation*}\int _0^1\frac {-\log f(x)}xdx=\frac {\pi ^2}{3ab},\end{equation*} 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.

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