极大右光滑扩张链
Maximal right smooth extension chains
摘要：
如果$w=uα$表示$α\in Σ=\{1,2\}$和$u\in Σ^*$，则称$w$为$u$的一个\textit{simple right extension}，用$u\prec w$表示。令$k$为正整数，$P^k(ε)$表示高度为$k$的所有$C^\infty$ -字的集合。设置$u_{1},\,u_{2},..., u_{m}\in P^{k}(ε)$，如果$u_{1}\prec u_{2}\prec ...\prec u_{m}$和没有$P^{k}(ε)$的元素$v$使得$v\prec u_{1}\text{or} u_{m}\prec v$，则$u_{1}\prec u_{2}\prec...\prec u_{m}$被称为高度为$k$的\textit{maximal right smooth extension (MRSE) chains}。本文证明了高度为$k$的\textit{MRSE}链构成高度为$k$的光滑字的一个划分，并给出了对每一正整数$k$，高度为$k$的\textit{MRSE}链的个数公式.此外，由于对于每个正整数$n$，存在长度为$n$的光滑字的最小高度$h_1$和最大高度$h_2$，我们发现高度为$h_1-1$和$h_2+1$的\textit{MRSE}链是分别用于建立长度为$n$的光滑字的个数的上界和下界的好的候选者，这比以前的方法更简单、更直观.
英文摘要：
If $w=uα$ for $α\in Σ=\{1,2\}$ and $u\in Σ^*$, then $w$ is said to be a \textit{simple right extension}of $u$ and denoted by $u\prec w$. Let $k$ be a positive integer and $P^k(ε)$ denote the set of all $C^\infty$-words of height $k$. Set $u_{1},\,u_{2},..., u_{m}\in P^{k}(ε)$, if $u_{1}\prec u_{2}\prec ...\prec u_{m}$ and there is no element $v$ of $P^{k}(ε)$ such that $v\prec u_{1}\text{or} u_{m}\prec v$, then $u_{1}\prec u_{2}\prec...\prec u_{m}$ is said to be a \textit{maximal right smooth extension (MRSE) chains}of height $k$. In this paper, we show that \textit{MRSE} chains of height $k$ constitutes a partition of smooth words of height $k$ and give the formula of the number of \textit{MRSE} chains of height $k$ for each positive integer $k$. Moreover, since there exist the minimal height $h_1$ and maximal height $h_2$ of smooth words of length $n$ for each positive integer $n$, we find that \textit{MRSE} chains of heights $h_1-1$ and $h_2+1$ are good candidates to be used to establish the lower and upper bounds of the number of smooth words of length $n$ respectively, which is simpler and more intuitionistic than the previous methods.
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