$b$的幂复合：渐近Enumeration和参数
Compositions into Powers of $b$: Asymptotic Enumeration and Parameters
摘要：
对于一个固定的整数基$b\geq2$，我们考虑的组合物的数量$1$到一个给定数量的功率的$b$，相关的，最大数量的表示一个正整数可以有一个有序的总和的功率的$b$。我们研究了这些数字的渐近增长，并给出精确的渐近公式，从而改善Molteni早期的结果。我们的方法使用生成函数，我们从无限的传递矩阵。用同样的技术，最大分母的分布和不同部分的数量进行了研究。
英文摘要：
For a fixed integer base $b\geq2$, we consider the number of compositions of $1$ into a given number of powers of $b$ and, related, the maximum number of representations a positive integer can have as an ordered sum of powers of $b$. We study the asymptotic growth of those numbers and give precise asymptotic formulae for them, thereby improving on earlier results of Molteni. Our approach uses generating functions, which we obtain from infinite transfer matrices. With the same techniques the distribution of the largest denominator and the number of distinct parts are investigated.
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