Maximal right smooth extension chains

If $w=u\alpha$ for $\alpha\in \Sigma=\{1,2\}$ and $u\in \Sigma^*$, 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 $P^k(\epsilon)$ denote the set of all $C^\infty$-words of height $k$. Set $u_{1},\,u_{2},..., u_{m}\in P^{k}(\epsilon)$, if $u_{1}\prec u_{2}\prec ...\prec u_{m}$ and there is no element $v$ of $P^{k}(\epsilon)$ 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.