完全有根树的性质与随机生成串的长度分布性质之间的关系
Relations between the properties of a complete rooted tree and the properties of a distribution of lengths of randomly generated strings
摘要：
让我们将高度为$n$的完全$m$ -ary有根树图表示为$G$。在本文的范围内，我们证明了$G$的属性和字符串的长度分布的期望和方差之间的某些关系，产生如下：从一个空字符串开始，我们从字母表$\{ α_1, α_2, \dots α_m \}$中挑选一个随机符号，并将其附加到字符串，该过程继续，直到我们看到一行中的特定符号的$n$实例。考虑随机变量$ξ_{m,n}$，其表示根据所描述的过程生成的字符串的长度。期望$\mathbb{E}[ξ_{m,n}]$和方差$\mathrm{Var}[ξ_{m,n}]$取决于$m$（字母表的大小）和$n$（定义字符串生成过程的停止标准的参数）。此外，让我们将$G$的所有2元组节点上的公共路径长度之和表示为$S_{m,n}$，并让我们将$G$中的边的总数表示为$T_{m,n}$。在本文的范围内，我们证明了以下关系对所有的$m,n \geq 1$都成立：$\mathbb{E}[ξ_{m,n}] = T_{m,n}$和$\mathrm{Var}[ξ_{m,n}] = (m-1) \cdot S_{m,n}$。虽然已知$\mathbb{E}[ξ_{2,n}]$和$T_{2,n}$均由来自On-Line Encyclopedia of Sequences（OEIS）的序列A000918描述，并且已知$S_{2,n}$由OEIS序列A286778描述，但我们证明了对A286778的新解释：该序列描述了$\mathrm{Var}[ξ_{2,n}]$ -直到我们看到$n$正面成一行为止，投掷公平硬币的次数的变化。
英文摘要：
Let's denote a complete $m$-ary rooted tree graph of height $n$ as $G$. In scope of this paper we prove the certain relations between the properties of $G$ and the expectation and variance of the distribution of lengths of strings, generated as follows: starting from an empty string we pick a random symbol from the alphabet $\{ α_1, α_2, \dots α_m \}$ and append it to the string, the process continues until we see $n$ instances of a specific symbol in a row. Consider a random variable $ξ_{m,n}$ that represents a length of a string generated according to the described process. The expectation $\mathbb{E}[ξ_{m,n}]$ and variance $\mathrm{Var}[ξ_{m,n}]$ depend on $m$ (the size of the alphabet) and $n$ (a parameter that defines a stopping criteria of the string generation process). Also, let's denote the sum of the common path length over all 2-tuples of nodes of $G$ as $S_{m,n}$, and let's denote the total number of edges in $G$ as $T_{m,n}$. In scope of this paper we prove that the following relations are true for all $m,n \geq 1$: $\mathbb{E}[ξ_{m,n}] = T_{m,n}$ and $\mathrm{Var}[ξ_{m,n}] = (m-1) \cdot S_{m,n}$. While it is known that both $\mathbb{E}[ξ_{2,n}]$ and $T_{2,n}$ are described by the sequence A000918 from the On-Line Encyclopedia of Integer Sequences (OEIS), and it is known that $S_{2,n}$ is described by the OEIS sequence A286778, we demonstrate a new interpretation for A286778: this sequence describes $\mathrm{Var}[ξ_{2,n}]$ - a variance of the number of tosses of a fair coin until we see $n$ heads in a row.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
