积分、划分和元胞自动机
Integrals, Partitions, and Cellular Automata
摘要：
证明了对于$0<a<b$，满足$f^a-f^b=x^a-x^b$的减函数为$$\int_0^1\frac{-\log f(x)}xdx=\frac{π^2}{3ab}$$.当$a$为整数且$b=a+1$为整数时，我们推出了几个组合结果.这些包括整数分区的数量没有$a$连续部分的渐近公式，和一类阈值增长的细胞自动机模型的自举渗流的亚稳态阈值的公式。
英文摘要：
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.
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