Maximal right smooth extension chains

If $w=u\alpha$ for $\alpha \in \mathrm{\Sigma }=\{1,2\}$ and $u\in {\mathrm{\Sigma }}^{\ast }$, then $w$ is said to be a $\mathit{\text{simple}}\mathit{\text{right}}\mathit{\text{extension}}$of $u$ and denoted by $u\prec w$\mathrm{.}Let$k$beapositiveintegerand$P^k(\epsilon)$denotethesetofall$C^\infty$-wordsofheight$k$\mathrm{.}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 $P^k(ϵ)$ 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 $\mathit{\text{MRSE}}$ chains of height $k$ constitutes a partition of smooth words of height $k$ and give the formula of the number of $\mathit{\text{MRSE}}$ 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 $\mathit{\text{MRSE}}$ 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.